# Geometry. Kellenberg Memorial High School

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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. Non-Collinear 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 well-defined 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 one-to-one 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 non-congruent 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 1-6 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 non-adjacent interior angles. As a result, that exterior angle must be greater than either non-adjacent 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 orean-theorem 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 = (5x-35), 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 counter-clockwise 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 x-axis is a horizontal number line and the y-axis 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 x-axis and the ordinate tells you to move up or down along the y-axis. 81

83 You try: 1. If a point is on the x-axis, what is the value of its ordinate? 2. What is the value of the abscissa of every point which is on the y-axis? 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 (x-values) 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 (y-values). 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 (y-values) are the same, the length of line segment CD can be found by taking the absolute value of the difference of the abscissas (x-values). 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 x-values and then take the average of the y-values. 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 y-intercept where the line intersects the y-axis 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 y-intercept. 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 (slope-intercept) 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 y-intercept 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 x-axis where the vertical line intersects the x-axis. 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 y-intercept 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 y-intercept: 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

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