Geometry. Kellenberg Memorial High School


 Phebe Bruce
 1 years ago
 Views:
Transcription
1 Geometry Kellenberg Memorial High School
2 Undefined Terms and Basic Definitions 1 Click here for Chapter 1 Student Notes
3 Section 1 Undefined Terms 1.1: Undefined Terms (we accept these as true) The word Geometry comes from two Ancient Greek words: ge metron meaning Earth meaning measure 1. Set: a set is a collection or group of objects with some common characteristic. Ex: The set of all students in your Geometry class, the set of all odd numbers, or the set of all cars made by Ford are examples. 2. Point: a point is basically a location in space. It can be represented by a dot on a piece of paper, named with a 2
4 capital letter. Ex: P or we can name it with a lower case letter point P Ex: line k Points have no size at all: no length, width, or thickness. Points indicate position or location as seen when graphing A line does not have a measurable length because it is infinitely points on the coordinate plane. long. 3. Line: A line is an infinite set of points. When we represent a 4. Plane: a plane is a set of points that forms a completely flat line, arrows are placed on each end of the line to illustrate that surface which extends infinitely in all directions. Think of it as the line extends infinitely in both directions. the world s biggest, flattest, thinnest piece of paper. (The symbol for Infinity is ). That infinite set of points, at least as far as we re concerned, In fact, a plane is so thin that it has no thickness at all. Think of usually forms a straight line. (We ll talk about curved lines much a textbook: if you pile 800 pages, one on top of another, you ll later in the course.) have a book that s an inch or two thick. But, To represent a line, we choose any 2 of the points on the line because planes have no thickness at all, and place an you can pile 8,000 of them, one on top of Ex: over them. line EF the other, and the thickness won t increase. 3
5 A plane is named by a single letter: plane N You try: Determine which undefined term describes the following. 4
6 Section 2 Basic Definitions 1.2 BASIC DEFINITIONS 5. Line Segment: a set of points consisting of two points on a line, called endpoints and the set of all points on the line between the endpoints. We can name a segment by placing a bar over the endpoints. ***A Line is named by any 2 points on the line, while a Line Segment is always named by its endpoints. 6. Ray: the set of all points in a half line, including the dividing point, which is called the endpoint of the ray. 5
7 A ray is named by placing an arrow pointing to the right over the angle. The vertex of the angle pictured above is at A. two capital letters We have a couple of options when it comes to naming an * 1st Letter Names the endpoint of the ray angle. We can use one letter: the vertex angle, and call the * 2nd Letter Names some other point on the ray angle above Angle A. Or we can use three letters as long as the one in the middle is EX: the vertex. So the picture above could be called CAB or BAC. Or, if we choose, we can name our angle using numbers or lower case letters. Examples: 7. Angle: the union of two rays having the same endpoint. Its symbol is either or Vertex (of the angle): the endpoints of each ray, or the corner of 6
8 We measure angles by determining the number of DEGREES c) Obtuse: 90 < θ < 180 contained in each one. An obtuse angle is one measuring greater than 90 and less than 180 degrees. It can look kind of like this: What are the different types of angles? As you probably remember from elementary school, there are a number of different types of angles, classified by the number of d) Straight: θ = 180 degrees it contains: A straight angle is an angle of exactly 180 degrees. a) Acute: 0 < θ < 90 An acute angle is one measuring greater than 0 and less than 90 degrees. It can look kind of like this: e) Reflex: 180 < θ < 360 A reflex angle is the one of these angle types you ve probably b) Right: θ = 90 never heard of. It s an angle whose measure is more than 180 A right angle is one measuring exactly 90 degrees. and less than 360 degrees It s the kind of angle found in a square or a rectangle. A right The problem, of course, is that reflex angles look just like acute angle is symbolized by a little box at the vertex, like this: angles: 7
9 The reflex angle in the picture above isn t what catches your 60 smaller units called, predictably enough, SECONDS. eye; it s the acute angle next to it that you tend to see. As a result, on those rare occasions this year when we want to talk about the reflex angle, we ll be sure to specify it. We can also Here are the symbols used for each of the units of measurement: Degrees Minutes Seconds mark the above diagram showing the reflex angle. Remember: θ is just another symbol used like the variables x or y, but is usually used with angles. It comes from the Greek alphabet: 1 Degree = 60 minutes (60 ) 1 Minute = 60 seconds (60 ) θ (pronounced theta ) So that means, for example, that Measuring Angles As you re already aware, angles can be measured in degrees. But sometimes, a single degree is too wide a measurement for a particular situation. Sometimes, we need a part of an angle in order to provide greater precision. Each angle can be broken down into 60 smaller units called MINUTES. And each minute, in turn, can be broken down into ¼ = 15 (since ¼ of 60 is 15) ½ = 30 (since ½ of 60 is 30) ¾ = 45 (since ¾ of 60 is 45) 30 minutes or 30 seconds act like the.5 in a decimal, for rounding purposes. That means that 30 minutes or 30 seconds is your Round UP number any number smaller will round DOWN. 8
10 Example 1: Round to the nearest degree Round to the nearest degree You try: Round the angle measure to the nearest minute. Example 2: Round to the nearest minute You try: Round the angle measure to the nearest degree Round to the nearest minute
11 8. Congruence: means having same length or measure (think: same size & shape) The symbol for congruence is: How would you mark line segments to show they are not It combines the equal sign: = (same size ) with the symbol congruent? for similarity ~ (same shape). We will learn more about similarity later in the year. b) Congruent Angles are angles which have the same measure. a) Congruent Segments are segments that have the same length. 10
12 9. Collinear Points are points that lie on the same straight 13. Parallel lines: straight lines that never intersect. The line. symbol for parallel is ll. For example we can say AB CD. 10. NonCollinear points, on the other hand, are points that DO NOT lie on the same straight line. Note that, unlike many of the definitions we ve seen thus far, 11. Midpoint is the point on a line segment that divides the Parallel addresses the DIRECTION a line goes, and not its segment into 2 segments. length. Two segments can certainly be parallel without being congruent. 14. Perpendicular lines: straight lines that intersect and form 12. Bisection of a Line Segment: a segment is bisected at a right angles (90 ). point if the point is the midpoint of the line segment. The symbol for perpendicular lines is:. So we can write AB BC if they intersect and form a right angle at B: 11
13 15. Perpendicular Bisector is, as you might think, a line or and you get degrees, not inches. segment which does two things: it cuts the line segment in half & forms right angles. 17. Complementary Angles are two angles whose sum is 90. For example, an angle of 40 and one of 50 are complements. Likewise, 1 and 2 in the diagram below are complements: 16. Angle Bisector: divides an angle into 2 congruent angles. 18. Supplementary Angles are two angles whose sum is 180. An angle of 116, then, would be the supplement of an angle of 64, since their sum is 180. Please note: when an angle is bisected, it forms two congruent ANGLES. It does NOT mean that the sides of the angles are congruent. Think about it for a second cut degrees in half 12
14 You try: 19. Adjacent Angles are angles that share a vertex and a side, 1. Find the complement of 32 but have no interior points in common. (The word adjacent means next to. ) 2. Find the complement of Find the supplement of 58 Which angles x & y do represent adjacent angles? 4. Find the supplement of (we will place x & y in the diagrams) 5. Find the supplement of Find the supplement of
15 20. A Linear Pair are two angles that are both supplementary You try: If and adjacent. 4 = 60, find the other 3 angles. 21. Vertical Angles are formed by intersecting lines. In the diagram below, as are 2 and 1 and 3 are vertical angles, 4. Which other pairs of angles can be found in this diagram? It s important to remember that VERTICAL ANGLES ARE CONGRUENT. Which pairs add to 180? (How would you mark the angles in the following diagram?) The four angles add up to degrees. 14
16 You try: In the following diagram a) Name the pairs of vertical angles. b) Name the adjacent angles at A. c) Name a linear pair. 15
17 Section 3 More Practice 1.3 More Practice Use your definitions to answers these questions: 1. Describe the type of angles. a) b) 90 c) d) 180 e) 216 f) g) h) i) 163 j)
18 2. Round each angle to the nearest degree. a) Points A, B & C are collinear and in that order Use the diagram below to answer the following: b) c) d) a) AB = 10, BC = 4 THEN AC =? e) b) AB = 23, AC = 72 THEN BC =? 3. Round each angle to the nearest minute. a) b) c) c) AC = 156, BC = 91 THEN AB =? d) e)
19 5. Find the complement of the following angles: a) Add the angles, use the following diagram. b) 38 c) 44 d) e) Subtract the angles, use the following diagram. 6. Find the supplement of the following angles: a) 133 b) 45 c) 99 For # 9 12 All points are collinear. d) e) Add the line segments, using the following diagram. 18
20 10. Subtract the line segments, using the following diagram. 13. Find the complement and supplement of each of the following algebraic expressions: a) m b) (3y) c) (y + 20) d) (q 35) e) (3p + 56) 19
21 Section 4 Distance and Absolute Value 1.4 Distance & Review of Absolute Value Distance: Absolute Value: = = 20
22 = (2) = = Using the above number line find the distance between the following points = (12) (2) = = = = 11. C&F 12. D&F 13. G&H 14. H&F 15. C&E 16. E&G 17. C&H 18. H&E 19. C&D 20. D&H 21
23 Sets and Venn Diagrams 2 Click here for Chapter 2 Student Notes
24 Section 1 Definitions Involving Sets 2.1 Definitions Involving Sets A Set is a welldefined collection of objects or numbers. Ex: Even integers greater than 0 and less than 10 A = { 2, 4, 6, 8 } We can name a set by assigning a capital letter to it, as we did in the above example. Note, too, that the members, or ELEMENTS, of a set are listed (or tabulated) inside the braces. Elements are the members that are contained in a given set. They can be numbers, letters, symbols or any other type of object. If we want to say that a particular number or object is an element of a set, we can use the symbol 23
25 For example: Set B is the set of common household pets B = {cat, dog, bird, hamster, fish, snake} Cat is an element of the set of common pets Cat B cat is an element in set B * When an element is not part of the set: Elephant B elephant is not an element in set B 24
26 Section 2 Kinds of Sets 2.2 Kinds of Sets A Finite Set is a set whose elements can be counted. In other words, there is a definite number of elements in the given set. Finite sets don t have to be EASY to count they can have millions of elements but there has to be a finite, or limited, number of elements in the set. Ex: the set of letters in the alphabet Ex: {1,2,3,, 200} An Infinite Set is a set whose elements cannot be counted. Ex: the set of real numbers Ex: {1,2,3, } 25
27 ***Please Note: {1, 2, 3, 4,..., 200} Examples of sets that are empty:  This means that the set includes The set of days that begin with the letter Z. the # s 1 to 200, consecutively. We use the three dots to Or represent the rest of the # s between 4 and 200. The months of the year that have 53 days. (it would take too long to write them all) Please note that we do NOT put the symbol into braces. {1, 2, 3,...}  This means that the set is an infinite set. The list of the elements in the set keeps going on forever. Doing so would mean that they were no longer empty. Nor do we put the number zero in braces. That would mean the set is a finite set containing the element 0. There s one more type of set we need to discuss: the Empty Set (or Null Set). It s empty because it s a set that has no elements. We represent the empty set one of two ways: The empty set is a subset* of every set. either the symbol or a pair of empty braces { }. 26
28 Section 3 Relationships Between Sets 2.3 Relationships Between Sets Equal Sets are sets that contain exactly the same elements though the order of elements does not matter. The symbol for equal sets is = Ex 1: A = {2, 4, 6, 8} B = {even counting numbers < 10} A=B Ex 2: C = { $, # } D = { $, &, } C=D 27
29 Ex 3: E = { x, y, z } The Universal Set is the entire set of elements under F = { a, b, c } consideration in a given situation. E F The Universal Set is represented by the symbol Sets E and F are not EQUAL sets, but they are an For example, if we wanted to examine the English Alphabet we example of EQUIVALENT sets. would state: Equivalent Sets are sets that have the same number of = {the letters of the alphabet} We can then look at the subsets* that are found in elements. Each element in one set can be matched to an V = {vowels} and C = {consonants} element in another. This is another way of saying that there is onetoone correspondence between the two sets an idea A Subset* is a set that contains the elements of another set. that will become important as you go on in math. Logically, then, every set is a subset of itself. The symbol for subsets is The symbol for equivalent sets is Ex: ~ G = {Beth, Mary, Kelly, Sue} H = {William, Joe, Mark, George} G~H Ex: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {3, 4, 5, 6} We can say that set B is a subset of set A B A Could set A be a subset of set B? 28
30 You try: Disjoint Sets are when sets do not have any common Rewrite these true statements using the symbol: elements  when their intersection yields the empty set. 1. Set A is a subset of itself. Ex: C = {2, 4, 6, 8} D = {1, 3, 5, 7} 2. Set B is a subset of itself. C 3. The empty set is a subset of A. Union of Sets is the combination of the elements of 2 or more D = { } or sets. 4. The empty set is a subset of B. The symbol for Union is. It will be easy to remember, since it looks like the capital letter U. Intersection of Sets: Sets intersect when they share at least Ex 1: A = {1, 2, 3} one common element. That intersection contains any overlap B = {4, 5, 6} between the sets. A B = {1, 2, 3, 4, 5, 6} The symbol for intersection is Ex: A= {1, 2, 3, 4, 5} A Ex2: C = {5, 6, 7, 8} B= {2, 4, 6, 8, 10} D = {4, 5, 6, 7, 9} B = {2, 4} C D = {4, 5, 6, 7, 8, 9} 29
31 Remember to copy each element only once. The Complement of a Set is the set of elements in the Universal Set that are NOT in the given set. Obviously, in order A good tip: Putting the elements into alphabetical/ to find the Complement of a set, you need to also know which numerical order is a helpful way to check that you have elements are in the Universal Set. listed all the elements needed. The symbol for the Complement is an apostrophe placed after the name of the set. Using the Empty Set with UNION & INTERSECTION Ex: ANY SET Ø = ANY SET ANY SET Ø = Ø = {1, 2, 3, 4, 5} For example if set A = {1, 2, 3} then A Ø = A or {1, 2, 3} A Ø = Ø 30
32 Section 4 The Venn Diagram 2.4 The Venn Diagram A Venn Diagram is a pictorial representation of a set. Venn Diagrams sometimes make it easier to find the answers to problems involving sets. In using Venn Diagrams, a rectangle will represent the Universal Set. It surrounds the rest of the diagram. Circles represent subsets of the Universal Set and the elements of a set are placed in the circle. 31
33 f) List A g) List B h) List (A B) i) List (A B) a) List j) List A b) List A B c) List B d) List A B e) List A B 32
34 d) List C e) List A a) List B f) List B C g) List A C b) List A c) List B h) List A B i) List A 33
35 j) List C p) List [ A B C ] B C k) List (A B) q) List A l) List (A C) r) List [ A m) List A C n) List A B o) List A B B C ] C 34
36 Introduction to Triangles 3 Click here for Chapter 3 Student Notes
37 Section 1 Definition of a Triangle and its Classifications 3.1 Definition of a Triangle and its Classifications Definition: A triangle is a 3 sided polygon. A polygon is a closed figure which is the union of line segments. (We will study more about polygons in chapter 5) Because a triangle has 3 sides it also has 3 interior angles. These three angles always add to 180. Labeling a triangle: Capital letters are used for the vertices. The same letters in lower case are used to represent the sides opposite those vertices. 36
38 Angles of the triangle are written using the single vertex letter 3. A RIGHT TRIANGLE has one right angle. or with three letters. (Again, why only one?) Sides can also be written by their line segment name. A triangle can be classified by its angles: 1. An ACUTE TRIANGLE has 3 acute We can also classify a triangle based on the number of angles. congruent sides it has. 2. An OBTUSE TRIANGLE has one obtuse angle. (Why only one?) 37
39 Classifying a triangle by its sides: 3. An EQUILATERAL TRIANGLE has three congruent sides and three congruent angles. 1. A SCALENE TRIANGLE has no congruent sides & therefore no congruent angles. Draw an example. Mark the sides and angles accordingly. Draw examples. Mark the sides and angles accordingly. What is the measure of each angle of the equilateral triangle? Why? 2. An ISOSCELES TRIANGLE has two congruent sides and two congruent angles. This triangle can also be called an EQUIANGULAR triangle, since the 3 angles are congruent. Draw examples. Mark the sides and angles accordingly. 38
40 You try: Some triangles have names for particular parts. Let s first EX 1: Classify a triangle with angles of 40, 60 and 80. discuss the right triangle: The easiest part of a right triangle to spot is its right angle. EX2: Classify a triangle with angles of 120, 30 and 30 It is symbolized with the small box in the right angle. The side across from the right angle is known as the EX 3: Classify a triangle with angles of 25, 90 and 65. HYPOTENUSE. It is always the longest side of the right triangle. EX 4: Classify a triangle with angles of 100, 60 and 20. The remaining two sides are the LEGS. They are always perpendicular to each other forming the right angle. EX 5: Classify a triangle with angles of and Label the parts of the right triangle below if AC CB. EX 6: Classify a triangle with angles of 90 and 45. EX 7: Classify a triangle with angles of 60 and Since C is the right angle, the other two angles of a right triangle must be acute. Why? What angle pair name can you give these two angles? 39
41 Now let s look at the Isosceles Triangle. As was stated earlier, You try: it has 2 congruent sides. Those congruent sides are called the 1. If the vertex angle is 106, what is the vertex angle? LEGS; the noncongruent side is the BASE. The two angles that share the base are called the BASE ANGLES. These angles are congruent. The angle formed by the legs is the VERTEX ANGLE. 2. If a base angle is 68, what is each base angle? Label the parts of the Isosceles Triangle below, if AB AC. 40
42 Section 2 Interior Angles of a Triangle 3.2 Interior Angles of a Triangle The sum of the 3 interior angles of any triangle is 180 degrees A triangle can have only 1 right angle A triangle can have only 1 obtuse angle If the triangle is a right triangle, then the remaining two angles must add up to the remaining 90 degrees. In other words, the acute angles of a right triangle are complements. If the triangle is equilateral, then it s also equiangular. (Remember, as stated earlier, that a triangle always has as many congruent angles as sides.) As a result, each angle of an equilateral triangle measures 60 degrees. (180 /3 = 60 ) If the triangle is an isosceles right triangle then each acute base angle, measures
43 The sum of the angles of a quadrilateral is 360 degrees. You try: The logic is simple: take any quadrilateral and draw a diagonal. The quadrilateral is now a pair of triangles, each having 1. Two angles of a triangle are 78 and 45. What is the 180 degrees. measure of the third angle? Then classify the triangle. 2. Two angles of a triangle are and What is the measure of the third angle? Then classify the triangle. 3. The angles of a triangle are represented by (3x + 1), (4x  12) and (7x + 9). Solve for x, find the measure of each angle and then classify the triangle. 42
44 Section 3 Exterior Angles of a Triangle 3.3 Exterior Angles of a Triangle When one side of a triangle is extended, the angle between that extension and the adjacent side is known as an Exterior Angle. (Remember, exterior means outside. ) In the diagram above, 1 is an exterior angle. The measure of an exterior angle of a triangle equals the sum of the two angles inside the triangle that are NOT adjacent to it the two interior angles that don t share a side with the exterior angle. 43
45 In the diagram above, that makes m 1 = m 2 + m 3. Below, draw examples of an obtuse triangle and a right triangle with their interior and exterior angles. What is the measure of exterior 1 if 2 = 46 and 3 = 77? What can be concluded about the sum of the exterior angles of any triangle? How many exterior angles does a triangle have? What are the degree measures of the exterior angles at A and C in the above diagram? Draw and label the angles. 44
46 You try: 3. Find the measure of a base of an isosceles if the exterior angle at the vertex measures Solve for x and find the measure of the angles Q & M. 4. In RST, angle S is a right angle and the m T = 38. Find the measure of the exterior angle at R. 2. Find the measure of the vertex of an isosceles if either of the exterior angles formed by extending the base measures
47 Section 4 Line Segments Associated with the Triangle 3.4 Line Segments Associated with the Triangle There are three types of line segments that exist in the triangle. 1. A MEDIAN is a line segment that is drawn from a vertex to the midpoint of the opposite side. In ABC, above, if M is the midpoint of BC, then AM is a median. We can also draw medians to sides AB and AC, once we locate their midpoints. 46
48 2. An ALTITUDE is a line segment drawn perpendicularly from How do you know which you re dealing with? The problem has a vertex to the opposite side, forming right angles. to tell you, either directly (using the words altitude or median or angle bisector ) or indirectly, by giving you the information that permits you to draw the correct conclusion. You try: For # s 16 Describe line segment DF in each of the following triangles. How many altitudes does a triangle have? 3. An ANGLE BISECTOR in a triangle does the same job it does when it s not in a triangle: it cuts an angle of the triangle into two congruent angles. 47
49 There are times when a single line segment can perform two or even all three of these jobs. In which triangle(s) can this occur? Illustrate below: 48
50 Section 5 Triangle Inequalities Triangle Inequalities Let s say your older brother has his driver s license and drives you to school each day. In the diagram above, let s let H symbolize your home. C can be your brother s favorite source of coffee, and K, of course, is Kellenberg. No matter how many shortcuts he knows, how fast a driver he is, how early he gets started, it s a basic fact of geometry that a detour for coffee on the way to school will add more mileage to the car than going straight to school the shortest distance between two points is a straight line. 49
51 As a result, we can make the following statement: the sum of two sides of a triangle is greater than the third side. You try: 1. Can these sets of numbers be the sides of a triangle? In particular, the sum of the lengths of two shortest sides must be greater than the longest side. a) {3, 4, 5} b) {11, 6, 9} c) {2, 8, 10} d) {.5, 12, 12} e) {7, 7, 7} f) {13, 30, 13} g) {6¼, 4½, 11} h) {1, 1, 3} i) {15, 8, 17} As far as angles in a triangle go, recall that the exterior angle of a triangle is equal to the sum of the nonadjacent interior angles. As a result, that exterior angle must be greater than either nonadjacent interior angle. (Think about it for a second if you have to add two interior angles to get the exterior, then that exterior MUST be greater than either of the two angles you added.) 50
52 Section 6 Lengths of Line Segments within Triangles 3.6 Lengths of Line Segments within Triangles Some curious things happen when we draw lines within triangles. The first happens when we connect midpoints. Here s the rule: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. In the diagram above, D is the midpoint of AB and E is the midpoint of AC. As a result, DE is parallel to BC, and half its length. 51
53 The second rule is this: The median to the hypotenuse of a right triangle is half the length of the hypotenuse. You try: For # 1 & 2 use this information: In triangle RST, QP joins the midpoints of sides RS and TS, respectively. 1. Find the length of QP if RT is: In the diagram above, BD is the median to hypotenuse AC. BD is half the length of AC. a) 14 b) d) e) 31½ 6x 17 c) 26.5 f) 8¾ c) f) (x+3) 2. Find the length of RT if QP is: a) 9 b) 13 d) 6¾ e) 29½ 52
54 Section 7 Angle  Side Relationship in a Triangle 3.7 Angle  Side Relationship in a Triangle Here are more interesting facts about the triangle: 1. The longest side of a triangle is opposite the triangle s largest angle. Likewise, the largest angle will be opposite the triangle s longest side. This is clearly demonstrated in the Right Triangle: The hypotenuse of the right triangle is the longest side of this triangle & it is opposite the 90 angle, the largest angle. 53
55 2. The shortest side of a triangle is opposite the triangle s smallest angle. Likewise, the smallest angle will be opposite the triangle s shortest side. You try: 1. In ABC, A = 50 & shortest sides of B = 60. Name the longest and ABC. 2. In ABC, AB = 11, BC = 10, and AC = 15. Name the largest and smallest angles of ABC. 54
56 Section 8 The Isosceles and Equilateral Triangles 3.8 The Isosceles and Equilateral Triangles The Isosceles and Equilateral triangles have some very useful properties. Let s go back to that basic rule: a triangle always has as many congruent angles as it has congruent sides. As a result, we have the rule: Base angles of an isosceles triangle are congruent. As you ll recall from section 3.1, the base angles are the angles touching the base. (The other angle is referred to as the vertex angle.) 55
57 The Converse (that s the reverse) of that rule is true as well: Another special property of the isosceles triangle is that the If two angles of a triangle are congruent, the sides opposite altitude is also the median is also the bisector of the vertex them are as well. angle; all three segments fall in the same place. (Note: very often, the converse of a true statement is NOT true; So, for example, in the diagram below, if we know that AD is an altitude, we know the following: this is one of the rare occasions when both are true.)  AD is perpendicular to BC. ( ADB and ADC are right angles.)  AD bisects BC (so BD DC)  AD bisects A (so BAD CAD) 56
58 The equilateral triangle also has some unique properties. Again going back to that basic rule, an equilateral triangle has 3 congruent angles. In other words, every equilateral triangle is also equiangular. (That means exactly what it sounds like: equal angles.) And since we know that a triangle has 180 degrees, we know that each angle of an equilateral triangle measures 60 degrees. Also, because an equilateral triangle is, by definition, also isosceles, all the properties we just discussed for the isosceles triangle also apply to the equilateral. 57
59 Section 9 The Pythagorean Theorem 3.9 The Pythagorean Theorem Pythagoras was a Greek philosopher and mathematician who lived in the 6th century BC. Among his many contributions to both fields is the theorem that bears his name: the Pythagorean Theorem. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Or, to put it simply: In a right triangle,, where a and b are the legs and c is the hypotenuse. That last bit is very important The a and b values are interchangeable, but c MUST be the hypotenuse. (Remember that the hypotenuse of a right triangle is the side across from the right angle.) 58
60 The Pythagorean Theorem can ONLY be used in a right For example, the world s most common Pythagorean Triple is triangle no other triangles have a hypotenuse. And it can only the triple. Do the math, and you ll see that be used to find the SIDES of a right triangle, never the angles. So if you re given a right triangle with legs of 3 and 4, you can simply state that the hypotenuse is 5, because it s a triple. Much of the time, in using Pythagorean Theorem, your answer (Note, it s not any 3 consecutive integers that will work; it s will be an irrational number the square root of a number that s these particular three.) not a perfect square. In such cases, please remember to leave your answer in simplest radical form. If you take a triple, and multiply each side by the same amount, you get another triple. So, for example, if you take that same triple, and multiply each side by 2, you get a triple. Pythagorean Triples three whole numbers that work in the Check and you ll see that it, too, works for the Pythagorean Pythagorean Theorem. Theorem. The most popular triples are, in order: Pythagorean Triples are common right triangles. If you have two of the three numbers in a triple, and they re in the correct positions, you can know the third number without doing the math
61 But they re not, by any stretch of the imagination, the only You try: triples that exist. Here s a list of a few more: 1. Find the hypotenuse of a right triangle, if the legs are: a) 9 & 12 b) 2&3 c) 5 & 6 d) 1.5 & 2 For a great take on the Pythagorean Theorem and what it DOESN T say, take a look at what happened when the Scarecrow from the Wizard of Oz was granted a brain: 2. Find the other leg when the hypotenuse and one leg is oreantheorem given: a) 26 & 10 b) 8 & 4 c) 17 & 3 d) 50 & In an isosceles right triangle, what are the measures of the legs if the hypotenuse is 10? 60
62 Section 10 Special Right Triangles 3.10 Special Right Triangles The first is derived from the Equilateral Triangle. It is the degree triangle. Here are some very special rules:  The shorter leg is half the hypotenuse.  The longer leg equals the shorter leg times 3. (Remember, of course, that the short leg is opposite the 30 degree angle, the long leg is opposite the 60, and the hypotenuse is opposite the right angle.) 61
63 Sometimes you will see the rule shown this way: The second special triangle is the degree triangle, or the Isosceles Right Triangle. In that triangle, the following rules apply:  The legs are congruent.  The hypotenuse equals the leg times 2. Examples: ***We will also investigate what happens when the side opposite the 60 degree angle is whole number.*** 62
64 When the hypotenuse of the Isosceles Right Triangle is a You try: (remember to draw pictures for each when solving) whole number then this rule applies: triangle  1. Find the remaining two sides when the hypotenuse is 12. A leg equals the hypotenuse times 2/2. 2. Find the remaining two sides when the side opposite the 30 degree angle is Find the remaining two sides when the side opposite the 60 degree angle is. ***However, if you forget these rules for the Isosceles Right Triangle, you can always use the Pythagorean Theorem to find 4. Find the remaining two sides when the hypotenuse the lengths of the legs or hypotenuse. is.* 63
65 You try: (remember to draw pictures for each when solving) triangle 1. Find the remaining two sides when the hypotenuse is. 2. Find the remaining two sides when one leg is Find the remaining two sides when the hypotenuse is Find the remaining two sides when one leg is. 64
66 Parallel Lines 4 Click here for Chapter 4 Student Notes
67 Section 1 Properties of Parallel Lines 4.1 Properties of Parallel Lines Parallel lines are two or more straight lines that do NOT intersect. You re familiar with the old example of railroad tracks as being parallel; if they weren t, the wheels of the train wouldn t be able to stay on the tracks. The symbol for parallel is. (That s convenient both because it shows you what it s symbolizing, and because it s contained within the word parallel. ) For example, we might write p q to describe the lines below: *Mark the above diagram to show lines p & q are 66
68 A Transversal is any line that intersects (cuts) 2 lines at  2 different points. opposite sides of the transversal.  Alternate Interior Angles: The pairs of interior angles on Alternate Exterior Angles: The pairs of exterior angles on opposite sides of the transversal. Line c is the transversal. same side of the transversal. They are supplementary.  When a transversal intersects any 2 lines, it creates 8 types of Same Side Interior Angles: The interior angles on the Same Side Exterior Angles: The exterior angles on the same side of the transversal. They are supplementary. angles (not including the linear pairs that exist). We will discuss these types of angles when a transversal intersects any 2 lines.  Interior Angles: The angles between the lines  Exterior Angles: The angles NOT between the lines  Vertical Angles: Angles defined earlier in Chapter 1  Corresponding Angles: The pairs of angles that are In the diagram below, we can identify the following angles. a b with transversal c in the same matching position 67
69 Interior Angles: Exterior Angles: Vertical Angles: Corresponding Angles: Alternate Interior Angles: Alternate Exterior Angles: Same Side Interior Angles: Same Side Exterior Angles: 68
70 Section 2 Exercises with Parallel Lines 4.2 Exercises with Parallel Lines You Try: (not all diagrams are drawn to scale) 1. Given the diagram m n with transversal k If m 1 = 127, fill in the remaining angles. 69
71 2. Given the diagram, e f with transversal g. If m 7 = 32 24, fill in the remaining angles. 4. What happens when there are 2 transversals? Given: m n with transversals p & q, m 1 = 54 and m 13 = 126. Find all of the missing angles in the diagram. 3. Given the diagram, p q with transversal r. If m 6 = , fill in the remaining angles. 70
72 Applying Algebra to Parallel Lines You try: For #1 & #2 AB // CD with transversal EF If the m 1 = 3x + 30 and the m measure of 1) m EGA = 2x and m GHC = 5x 54. Find: a) x = 8 = x + 60, find the 3. b) m EGA = c) m EGB = The steps: 1. What type of angles? 2. Relationship of angles 3. Set up Equation 4. Solve equation 5. Substitute 71
73 2) m Find: AGH = 3x  40 and m a) x = CHG = x Given: g h with transversals w & z, r g, m 7 = (5x35), m 16 = (x+14) and m 17= (4x+1). Solve for x & then find all of the missing angles in the diagram. b) m AGH = c) m CHG = d) m BGH = 72
74 You try: (how well do you know the angles?) Use the following diagram: a // b cut by transversal c. Write the types of angles in the spaces provided. a. 2 and 7 are called angles. b. 1 and 5 are called angles. c. 4 and 1 are called angles. d. 3 and 5 are called angles. e. 2 and 8 are called angles. f. 4 and 5 are called angles. g. 6 and 8 is an example of a(n). 73
75 Polygons 5 Click here for Chapter 5 Student Notes
76 Section 1 Polygons 5.1 Polygons A POLYGON is a closed figure which is the union of line segments. Polygons have sides and corners. Those corners are called VERTICES. (A single one is called a VERTEX.) We can classify polygons according to the number of sides they have: 75
77 When we re labeling a polygon, we choose one vertex as a starting place. Then we go from one vertex to the next, either in clockwise or counterclockwise order it doesn t matter which, but it is important that the vertices be labeled in order. CONSECUTIVE VERTICES of a polygon are vertices that share a side. So, for example, in the diagram below of quadrilateral ABCD, A & B, B & C, C & D and D & A are all consecutive vertices. CONSECUTIVE SIDES are, predictably enough, sides that share a common vertex. Using the same quadrilateral, AB & BC, BC & CD, CD & DA and DA &AB are all consecutive sides. 76
78 Section 2 The Interior and Exterior Angle of Polygons 5.2 The Interior and Exterior Angles of Polygons We will explore the angles of various polygons and develop the formulas needed. Summary of Formulas: 77
79 Section 3 The Regular Polygon Chart 5.3 The Regular Polygon Chart 78
80 Coordinate Geometry 6 Click here for Chapter 6 Student Notes
81 Section 1 Plotting Points & the Coordinate Plane 6.1 Plotting Points & the Coordinate Plane If you ve ever used MapQuest or Google Earth, you know that every location has its own unique address. Those addresses are actually based on the idea of graphing points in the coordinate plane. The Coordinate Plane, also known as the Cartesian or Rectangular Plane, is made up of two perpendicular lines called axes. The xaxis is a horizontal number line and the yaxis is a vertical number line. Every point on the plane can be located by its coordinates: the ordered pair made up of its x coordinate or abscissa followed by its y coordinate or ordinate. A point is written as (x, y). 80
82 The x and y axes divide the plane into 4 Quadrants (or quarters) numbered as shown: Quadrants are numbered counter clockwise. Each quadrant has specific x and y values. When plotting a point, the abscissa tells you to move left or right along the xaxis and the ordinate tells you to move up or down along the yaxis. 81
83 You try: 1. If a point is on the xaxis, what is the value of its ordinate? 2. What is the value of the abscissa of every point which is on the yaxis? 3. What are the coordinates of the origin? 4. Tell the sign of the abscissa and the ordinate of a coordinate point if the point lies in quadrant: a) I b) II c) III d) IV 5. Tell which quadrant or axis the following points lie: a) (9, 10) c) (0, 15) e) (14, 0) g) (16, 24) b) (3, 11) d) (20, 18) f) (17, 13) h) (14, 29) 82
84 Section 2 Areas in Coordinate Geometry 6.2 Areas in Coordinate Geometry Once we plot a polygon on the coordinate plane, it s a fairly simple matter to find its area. If the polygon has horizontal and vertical line segments that represent sides and/or altitudes, it s really just a matter of counting boxes and using basic formulas. But if the polygon has slanted sides, the process is just a little more extensive. Here is an example and the steps to do so: 83
85 You try: 1. Plot the points and then find the area of each triangle. a) A(4,3) B(1, 3) C(2, 1) b) D(3, 3) E(4, 5) F(2, 4) c) T(3, 3) R(0, 5) I(4, 1) d) S(4, 3) K(1, 5) Y(6,2) e) P(5, 4) Q(5, 6) R(2, 5) f) K(3, 2) L(1, 8) M(3, 4) 2. Plot the points and then find the area of each quadrilateral. a) A(6, 4) B(3, 2) C(2, 3) D(9, 0) b) Q(0, 3) R(6, 1) S(2, 3) T(3, 1) c) E(3, 6) F(6, 2) G(2, 6) H(7, 1) d) J(5, 2) K(3, 5) L(2, 3) M(1, 1) e) P(1, 2) Q(2, 7) R(6, 10) S(9, 1) f) W(4,3) X(1,7) Y(6,6) Z(10,4) g) G(6, 7) H(2, 3) I(4, 4) J(4, 3) 84
86 3. Plot the points and find the area of the pentagon if the coordinates are A(0, 7) B(2, 8) C(6, 4) D(0, 0) and E(2, 3). 85
87 Section 3 Distance Between Two Points 6.3 Distance between Two Points When we want to find the LENGTH of a line segment, it s another way of saying we want to find the DISTANCE between the two endpoints. Points A (2, 5) and B (2, 1) form a vertical line segment. Since the abscissas (xvalues) are the same, the length of line segment AB can be found by taking the absolute value of the 86
88 difference of the ordinates (yvalues). Of course, not all line segments are horizontal or vertical. Let s look at points E (3, 4) and F (2,3). Points C (3, 3) and D (5,3) form a horizontal line segment. Since the ordinates (yvalues) are the same, the length of line segment CD can be found by taking the absolute value of the difference of the abscissas (xvalues). What is the length of line segment EF? 87
89 One way to find the length of line segment EF is to create However, there is another method to find the Distance between a right triangle and use the Pythagorean Theorem, since any 2 points or the Length of any line segment. EF will become the hypotenuse of the right triangle. Find the horizontal and vertical line segments lengths and then substitute the values into 88
90 You try: 1. Find the distance between points G (2, 5) and H (4, 3) 2. Points R (4, 4) and S (2, 3) form line segment RS. Find its length as a radical answer and as a decimal answer to the nearest tenth. 3. Find the distance between points (m, 0) and (0, p). 89
91 Section 4 Midpoint of a Line Segment 6.4 Midpoint of a Line Segment The midpoint of a line segment is exactly what it sounds like: the point in the middle of the two endpoints. To easily remember the formula: Take the average of the xvalues and then take the average of the yvalues. 90
92 Find the midpoint of the line of the line segment which joins the point R (2, 5) & the point S (4, 1) The midpoint is (3, 2) *Always remember that you want to express your final answer as a coordinate point, an ordered pair You try: 1. Find the midpoint of the following sets of point: a) (3, 8) & (5, 6) b) (2, 7) & (8, 10) Now use the distance formula to show that (3,2) is indeed c) (16, 9) & (0, 5) d) (17, 1) & (5, 6) the midpoint of RS. Label (3,2) as point M and show that e) (3, 12) & (3, 10) f) (k, 0) & (0, m) RM is congruent to SM. 2. In a circle, the diameter s endpoints are (4, 3) and (2, 9). Find the center of the circle. 91
93 3. Line segment RS has endpoint R (14, 6) and midpoint M (3, 2). Find the coordinates of endpoint S. 4. The midpoint of segment QR is (5, 2½). Find endpoint Q if endpoint R has coordinates (20, 7). 92
94 Section 5 Slope of a Line 6.5 Slope of a Line The slope of a line tells you something about the direction in which the line slants. (Slope indicates Direction.) The formula for slope is: In other words, it s the change in the y values (vertical movement) divided by the change in the x values (horizontal movement). It s important to note that the order in which you use the points can t change; if you use Point A first on top, then Point A must also be used first on the bottom. 93
95 Find the slope of line AB if points A (2, 5) and B (4, 5) Find the slope of line CD if points C (2, 4) and D (2, 3) are on the line. are on the line. Vertical lines will always have slopes that are undefined. Horizontal lines will always have slopes of ZERO. They are said to have NO SLOPE. 94
96 Find the slope of line EF if points E (2, 2) and F (5, 4) Find the slope of line GH if points G (5, 2) and H (0, 5) are on the line. are on the line. This is a line that has a POSITIVE SLOPE. This is a line that has a NEGATIVE SLOPE. It is a line that leans to the right. It is a line that leans to the left. 95
97 You try: 1. Determine the slope of a line formed by each set of points: a) (4, 7) & (8, 3) b) (3, 6) & (3, 2) c) (5, 4) & (2, 1) d) (2, 3) & (0, 3) e) (6, 2) & (2, 2) f) (b, a) & (0, a) 2. Find the missing value of y so that the line passing through the points (5, 3) and (5, y) has a slope of 1/2. 3. Find the missing value of x so that the line passing through the points (4, 1) and (x, 3) has a slope of 2/3. 4. Determine if the following points are collinear. a) (6, 8) (0, 5) (4, 4) b) (1, 8) (1, 2) (4, 8) 96
98 Section 6 Parallel and Perpendicualar Lines 6.6 Parallel and Perpendicular Lines As we ve been discussing, slope tells you something about the direction of a line. So it stands to reason that if two lines go in the same direction or are parallel they would have the same slope. 1. Plot line AB: A (3, 5) and B (1, 2) and then find its slope. 2. Plot line CD: C (2, 2) and D (2, 5) and then find its slope. 97
99 Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals. However, perpendicular lines intersect to form right angles. Their slopes have a special relationship than just any pair of intersecting lines. You try: 1. Plot line EF: E (3, 4) and F (1, 1) and then find its slope. 2. Plot line GH: G (2, 3) and H (3, 1) and then find its slope. 1. Determine if line EF is parallel or perpendicular to line GH? a) E (1, 6) F (5, 4) & G (1, 2) H (3, 0) b) E (2, 2) F (1, 5) & G (6, 8) H (0, 2) c) E (4, 1) F(4, 7) & G (3, 1) H (7, 1) 2. Plot triangle ABC with vertices A (6, 8), B (10, 2) and C (4, 4). Show that if you join the midpoints of AC and AB, that the line segment formed is both parallel to CB and is half its length. 98
100 Section 7 Coordinate Proofs 6.7 Coordinate Proofs Now that you re comfortable with slope, distance and midpoint, we can use them to prove figures in coordinate geometry. Remember: Slope means Direction and Distance means Length. Here s how a proof works: First, you plot the coordinates and label the lines or vertices. Second, you determine which formula(s) you ll need, depending on what you re asked to prove. Write the formula(s) out, at least the first time you use it. 99
101 Third, use the formula(s) as many times as necessary to prove what you re asked. Last, write a sentence or two, starting with the word Since to explain how the work you ve done proves what you ve been asked to prove. Triangles will be the first type of polygon to be used with coordinate proofs. (section 6.8) Quadrilaterals will be used after first learning about their properties. (section 7.9) 100
102 Section 8 Proving Triangles Using Coordinate Geometry 6.8 Proving Triangles Using Coordinate Geometry To Prove a Triangle is Isosceles:  Show that 2 sides are congruent (distance formula) To Prove a Triangle is a Right Triangle:  Show that there is 1 right angle sides by showing that 2 consecutive sides are perpendicular (slope formula)  Show that the Pythagorean Theorem works after finding the length of each side (distance formula & 101
103 To Prove a Triangle is a Right Isosceles Triangle  Show that 2 sides are congruent & that there is 1 right angle by showing that 2 consecutive sides are perpendicular (distance & slope formulas)  Show that 2 sides are congruent & then use the Pythagorean Theorem to show it is a right triangle (distance formula & 102
104 Section 9 Equation of a Line 6.9 Equation of a Line Any straight line can be expressed in the form of an equation. That equation is typically written in standard form: y = mx + b Where m = slope b = the yintercept where the line intersects the yaxis at the point (0,b) x & y = the coordinates of any point that is on the line Example: Write in standard form: 3x 2y + 8 = 0 Find the slope and yintercept. 103
105 Section 10 Writing the Equation of a Line 6.10 Writing the Equation of a Line Just as every equation can be graphed, we can also write the equation of any line we graph. We will be writing the linear equation in standard (slopeintercept) form. Here are the steps: First determine the slope if it is not given. If necessary use the formula for slope: 1. Write the standard form: y = mx + b 2. Substitute that value of m in y = mx + b 3. Substitute the x and the y, the values of the coordinates of a point on the line, in y = mx + b. Be careful to put them into 104
106 the right places. Ex3: Write a linear equation that passes through the points 4. Solve for b. (6, 4) and (8, 5). 5. Rewrite y = mx + b, replacing the values of m and b into the equation: y = mx + b Ex1: Write a linear equation with a slope of 3 and yintercept of 12. m = 3 b = 12 y = mx + b y = 3x + 12 substitute in m & b Ex2: Write a linear equation with a slope of 3 and passes through the point (2, 7). 105
107 Equation of a Horizontal Line: Since a horizontal line has a slope of Zero then y = mx + b y = 0x + b Equation of a Vertical Line: Since a vertical line has an undefined slope or No Slope then * # is the number on the xaxis where the vertical line intersects the xaxis. 106
108 Section 11 Linear Equation Practice Problems 6.11 Linear Equation Practice Problems 1. Write each equation in standard form. Find the slope and yintercept of each. a) y = 8 b) x 3 + y = 2 c) 12 x 3y = 9 d) ½x + y = 2 e) 4y + 2x 6 = 0 f) 2x + y + 7 = 0 2. Write a linear equation in standard form given the slope and yintercept: a) m = 1/3 b = 6 b) m = 2 b = 4/5 c) m = 1 b = 9 d) m = 1/2 b = 8 e) m = 6 b = 3 f) m = 2/3 b = 5 107
109 3. Write a linear equation in standard form given the slope and a point: a) m = 3 & (2, 7) b) m = 5 & (1, 3) c) m = ½ & (8, 3) d) m = 2 & (1, 4) e) m = 2/3 & (6, 5) g) m = 3 & (4, 2) 6. Write the equation of line that has a slope of 1/4 and passes through the point (0, 3). 7. Write the equation of a horizontal line that passes through f) m = 4/5 & (10, 1) the point (8, 3). h) m = 1/3 & (3, 9) i) m = 1 & (11, 3) 8. Write the equation of a vertical line that passes through the point (5, 9). 4. Write a linear equation given two points: a) (2, 3) (1, 1) b) (3, 6) (6, 8) c) (0, 3) (6, 0) d) (3, 5) (8, 5) e) (4, 7) (4, 9) f) (5, 2) (7, 8) 9. Given line: y = 3x 5 Write the equation of a line that is parallel to this line and has a y intercept of 7. New line s equation: 5. Which point(s) satisfy the given equation? a) y = 3x + 1 (4, 1) (3, 10) (2, 5) (0,1) b) y x = 3 (8,5) (2, 5) (3, 0) (0, 3) 108
Centroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationSOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
More information55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
More informationGeometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment
Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points
More informationFinal Review Geometry A Fall Semester
Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over
More informationUnit 3: Triangle Bisectors and Quadrilaterals
Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationGeometry Regents Review
Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationIntermediate Math Circles October 10, 2012 Geometry I: Angles
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationChapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More information0810ge. Geometry Regents Exam 0810
0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationGeometry Module 4 Unit 2 Practice Exam
Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationGEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!
GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA (x₂x₁)²+(y₂y₁)² Find the distance between the points ( 3,2) and
More information/27 Intro to Geometry Review
/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationGeometry Chapter 1 Vocabulary. coordinate  The real number that corresponds to a point on a line.
Chapter 1 Vocabulary coordinate  The real number that corresponds to a point on a line. point  Has no dimension. It is usually represented by a small dot. bisect  To divide into two congruent parts.
More informationGeometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C2 Vertical Angles Conjecture If two angles are vertical
More informationhttp://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4
of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the
More information56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.
6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which
More informationIsosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors
Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A
More informationThe Triangle and its Properties
THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three
More informationPOTENTIAL REASONS: Definition of Congruence:
Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name
More information2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?
MATH 206  Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of
More information5.1 Midsegment Theorem and Coordinate Proof
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle  A midsegment of a triangle is a segment that connects
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationA summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:
summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of midpoint and segment bisector M If a line intersects another line segment
More information1. absolute value : The distance from a point on the number line to zero Example:  4 = 4; 4 = 4
1. absolute value : The distance from a point on the number line to zero  4 = 4; 4 = 4 2. addition property of opposites : The property which states that the sum of a number and its opposite is zero 5
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More information100 Math Facts 6 th Grade
100 Math Facts 6 th Grade Name 1. SUM: What is the answer to an addition problem called? (N. 2.1) 2. DIFFERENCE: What is the answer to a subtraction problem called? (N. 2.1) 3. PRODUCT: What is the answer
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name
More informationGeometry, Final Review Packet
Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a
More informationGeometry in a Nutshell
Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where
More informationTopics Covered on Geometry Placement Exam
Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals
More informationGEOMETRY FINAL EXAM REVIEW
GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.
More informationMODERN APPLICATIONS OF PYTHAGORAS S THEOREM
UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of
More informationABC is the triangle with vertices at points A, B and C
Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry  symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the
More informationA convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.
hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your
More information65 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.
ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram
More informationUse the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.
Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than
More informationHow Do You Measure a Triangle? Examples
How Do You Measure a Triangle? Examples 1. A triangle is a threesided polygon. A polygon is a closed figure in a plane that is made up of segments called sides that intersect only at their endpoints,
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
More informationMaximizing Angle Counts for n Points in a Plane
Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationConjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)
Mathematical Sentence  a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement
More informationThis is a tentative schedule, date may change. Please be sure to write down homework assignments daily.
Mon Tue Wed Thu Fri Aug 26 Aug 27 Aug 28 Aug 29 Aug 30 Introductions, Expectations, Course Outline and Carnegie Review summer packet Topic: (11) Points, Lines, & Planes Topic: (12) Segment Measure Quiz
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationGrade 4  Module 4: Angle Measure and Plane Figures
Grade 4  Module 4: Angle Measure and Plane Figures Acute angle (angle with a measure of less than 90 degrees) Angle (union of two different rays sharing a common vertex) Complementary angles (two angles
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationPolygons are figures created from segments that do not intersect at any points other than their endpoints.
Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures created from segments that do not intersect at any points other than their endpoints. A polygon is convex if all of the interior angles
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationLines, Segments, Rays, and Angles
Line and Angle Review Thursday, July 11, 2013 10:22 PM Lines, Segments, Rays, and Angles Slide Notes Title Lines, Segment, Ray A line goes on forever, so we use an arrow on each side to indicate that.
More informationGRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?
GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationCongruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key
Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,
More informationMA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane.
MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane. Symmetry When you can fold a figure in half, with both sides congruent, the fold line
More informationA. 3y = 2x + 1. y = x + 3. y = x  3. D. 2y = 3x + 3
Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x  1 D. y = 3x + 1
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationQuadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid
Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More information104 Inscribed Angles. Find each measure. 1.
Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semicircle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More information2014 Chapter Competition Solutions
2014 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering
More informationUnit 6 Grade 7 Geometry
Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson
More informationMATH STUDENT BOOK. 8th Grade Unit 6
MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular
More informationGeometry: Euclidean. Through a given external point there is at most one line parallel to a
Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows : gradient = vertical horizontal horizontal A B vertical
More informationUnit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook
Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfdprep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationChapter 5: Relationships within Triangles
Name: Chapter 5: Relationships within Triangles Guided Notes Geometry Fall Semester CH. 5 Guided Notes, page 2 5.1 Midsegment Theorem and Coordinate Proof Term Definition Example midsegment of a triangle
More information