# 1.1 Identify Points, Lines, and Planes

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean. Point - A point has no dimension. It is represented by a dot. Line - A line has one dimension. It is represented by a line with two arrowheads, but it extends without end. Through any two points, there is exactly one line. You can use any two points on a line to name it. Line AB (written as ) and points A and B are used here to define the terms below. Plane - A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane. Collinear points - Collinear points are points that lie on the same line. Coplanar points - Coplanar points are points that lie in the same plane. Defined terms - In geometry, terms that can be described using known words such as point or line are called defined terms. Line segment, Endpoints - The line segment AB, or segment AB, (written as ) consists of the endpoints A and B and all points on that are between A and B. Note that can also be named. Ray - The ray AB (written as ) consists of the endpoint A and all points on that lie on the same side of A as B. Note that and are different rays. Opposite rays - If point C lies on between A and B, then and are opposite rays. Intersection - The intersection of the figures is the set of points the figures have in common. The intersection of two The intersection of two different lines is a point. different planes is a line. EXAMPLE 1 Name points, lines, and planes a. Give two other names for and for plane Z. b. Name three points that are collinear. Name four points that are coplanar.

2 EXAMPLE 2 Name segments, rays, and opposite rays a. Give another name for. b. Name all rays with endpoint W. Which of these rays are opposite rays? Solution EX (1.1 cont.) EXAMPLE 3 Sketch intersections of lines and planes a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point. EXAMPLE 4 Sketch intersections of planes Sketch two planes that intersect in a line. Solution STEP 1 Draw a vertical plane. Shade the plane. STEP 2 Draw a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. STEP 3 Draw the line of intersection.

3 1.1 Cont.

4 2.4 Use Postulates and Diagrams Obj.: Use postulates involving points, lines, and planes. Key Vocabulary Line perpendicular to a plane - A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. Postulate - In geometry, rules that are accepted without proof are called postulates or axioms. POSTULATES Point, Line, and Plane Postulates POSTULATE 5 - Through any two points there exists exactly one line. POSTULATE 6 - A line contains at least two points. POSTULATE 7 - If two lines intersect, then their intersection is exactly one point. POSTULATE 8 - Through any three noncollinear points there exists exactly one plane. POSTULATE 9 - A plane contains at least three noncollinear points. POSTULATE 10 - If two points lie in a plane, then the line containing them lies in the plane. POSTULATE 11 - If two planes intersect, then their intersection is a line. CONCEPT SUMMARY - Interpreting a Diagram When you interpret a diagram, you can only assume information about size or measure if it is marked. YOU CAN ASSUME YOU CANNOT ASSUME All points shown are coplanar. G, F, and E are collinear. AHB and BHD are a linear pair. BF and CE intersect. AHF and BHD are vertical angles. BF and CE do not intersect. A, H, J, and D are collinear. BHA CJA AD and BF intersect at H. AD BF or m AHB = 90⁰ EXAMPLE 1 Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. Solution

5 EXAMPLE 2 Identify postulates from a diagram Use the diagram to write examples of Postulates 9 and 11. (2.4 cont.) EXAMPLE 3 Use given information to sketch a diagram Sketch a diagram showing RS perpendicular to TV, intersecting at point X. EXAMPLE 4 Interpret a diagram in three dimensions

6 2.4 Cont.

7 1.2 Use Segments and Congruence Obj.: Use segment postulates to identify congruent segments. Key Vocabulary Postulate, axiom - In Geometry, a rule that is accepted without proof is called a postulate or axiom. Coordinate - The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. Distance - The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. Between- When three points are collinear, you can say that one point is between the other two. Congruent segments - Line segments that have the same length are called congruent segments. POSTULATE 1 Ruler Postulate POSTULATE 2 Segment Addition If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. Lengths are equal. AB = CD is equal to Segments are congruent. is congruent to EXAMPLE 1 Apply the Ruler Postulate Measure the length of CD to the nearest tenth of a centimeter. Solution EXAMPLE 2 Apply the Segment Addition Postulate MAPS The cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri. Solution Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate. LS = LT + TS = = 740 The distance from Lubbock to St. Louis is about 740 miles.

8 EXAMPLE 3 Find a length Use the diagram to find KL. Solution Use the Segment Addition Postulate to write an equation. Then solve the equation to find KL. (1.2 cont.) EXAMPLE 4 Compare segments for congruence Plot F(4, 5), G( 1, 5), H(3, 3), and J(3, 2) int a coordinate plane. Then determine whether FG and HJ are congruent.

9 1.3 Use Midpoint and Distance Formulas Obj.: Find lengths of segments in the coordinate plane. Key Vocabulary Midpoint - The midpoint of a segment is the point that divides the segment into two congruent segments. Segment bisector - A segment bisector is a point, ray, line, line segment, or plane the at intersects the segment at its midpoint. M is the midpoint of AB. CD is a segment bisector of AB.. So, AB AB and AM = MB. The Midpoint Formula KEY CONCEPT The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints. If A(x₁, y₁) and B(x₂, y₂) are points in a coordinate plane, then the midpoint M of AB has coordinates x 1 x y1 y 2 2, 2 2 The Distance Formula KEY CONCEPT If A(x₁, y₁) and B(x₂, y₂) are points is a coordinate plane, then the distance between A and B is 2 AB = x x y y 1 So, AB AB and AM = MB. EXAMPLE 1 Find segment lengths Find RS. EXAMPLE 2 Use algebra with segment lengths ALGEBRA Point C is the midpoint of BD. Find the length of BC.

10 EXAMPLE 3 Use the Midpoint Formula a. FIND MIDPOINT The endpoints of PR are P( 2, 5) and R(4, 3). Find the coordinates of the midpoint M. (1.3 cont.) b. FIND ENDPOINT The midpoint of AC is M(3, 4). One endpoint is A(1, 6). Find the coordinates of endpoint C. The Distance Formula is based on the Pythagorean Theorem, which you will see again when you work with right triangles in Chapter 7. Distance Formula Pythagorean Theorem EXAMPLE 4 Use the Distance Formula What is the approximate length of TR, with endpoints T( 4, 3) and R(3, 2)?

11 1.3 Cont.

12 1.4 Measure and Classify Angles Obj.: Name, measure, and classify angles. Key Vocabulary Angle - An angle consists of two different rays with the same endpoint. Sides, vertex of an angle - The rays are the sides of the angle. The endpoint is the vertex of the angle. Measure of an angle - A protractor can be used to approximate the measure of an angle. An angle is measured in units called degrees (⁰). Words The measure of WXZ is 32⁰. Symbols m WXZ = 32⁰ Congruent angles - Two angles are congruent angles if they have the same measure. Angle bisector - An angle bisector is a ray that divides an angle into two angles that are congruent. POSTULATE 3 - Protractor Postulate Consider OB and a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180. The measure of AOB is equal to the absolute value of the difference between the real numbers for OA and OB. CLASSIFYING ANGLES Angles can be classified as acute, right, obtuse, and straight, as shown below. the red square means a right angle POSTULATE 4 - Angle Addition Postulate (AAP) Words If P is in the interior of RST, then the measure of RST is equal to the sum of the measures of RSP and PST. Symbols If P is in the interior of RST, then m RST = m RSP + m PST. interior Angle measures are equal. Angles are congruent. the arcs show that the m A = m B A B angels are congruent equal to is congruent to

13 EXAMPLE 1 Name angles Name the three angles in the diagram. (1.4 cont.) EXAMPLE 2 Measure and classify angles Use the diagram to find the measure of the indicated angle. Then classify the angle. a. WSR b. TSW c. RST d. VST EXAMPLE 3 Find angle measures ALGEBRA Given that m GFJ = 155⁰, find m GFH and m HFJ. EXAMPLE 4 Identify congruent angles Identify all pairs of congruent angles in the diagram. If m P = 120⁰, what is m N? EXAMPLE 5 Double an angle measure In the diagram at the right, WY bisects XWZ, and m XWY = 29⁰. Find m XWZ

14 1.4 Cont.

15 2.6 Prove Statements about Segments and Angles Obj.: Write proofs using geometric theorems. Key Vocabulary Proof - A proof is a logical argument that shows a statement is true. There are several formats for proofs. Two-column proof - A two-column proof has numbered statements and corresponding reasons that show an argument in a logical order. Theorem - The reasons used in a proof can include definitions, properties, postulates, and theorems. A theorem is a statement that can be proven. THEOREMS Congruence of Segments Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment AB, AB AB. Symmetric If AB CD, then CD AB. Transitive If AB CD and CD EF, then AB EF. Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. Reflexive For any angle A, A A. Symmetric If A B, then B A. Transitive If A B and B C, then A C. EXAMPLE 1 Write a two-column proof Use the diagram to prove m 1 = m 4. Given: m 2 = m 3, m AXD = m AXC Prove: m 1 = m 4 EXAMPLE 2 Name the property shown Name the property illustrated by the statement. If 5 3, then 3 5

16 EXAMPLE 3 Use properties of equality If you know that BD bisects ABC, prove that m ABC is two times m 1. Given: BD bisects ABC. Prove: m ABC = 2 m 1 (2.6 cont.) EXAMPLE 4 Solve a multi-step problem Interstate There are two exits between rest areas on a stretch of interstate. The Rice exit is halfway between rest area A and Mason exit. The distance between rest area B and Mason exit is the same as the distance between rest area A and the Rice exit. Prove that the Mason exit is halfway between the Rice exit and rest area B. CONCEPT SUMMARY - Writing a Two-Column Proof In a proof, you make one statement at a time, until you reach the conclusion. Because you make statements based on facts, you are using deductive reasoning. Usually the first statement-andreason pair you write is given information. Proof of the Symmetric Property of Angle Congruence GIVEN 1 2 PROVE 2 1 Copy or draw diagrams and label given information to help develop proofs. The number of Remember to give a reason 1 2 statements will vary for the last statement.

17 Definitions, postulates Theorems Statements based on facts that you know or on conclusions from deductive reasoning 2.6 Cont.

18 1.5 Describe Angle Pair Relationships Obj.: Use special angle relationships to find angle measure. Key Vocabulary Complementary angles - Two angles are complementary angles if the sum of their measures is 90⁰. Supplementary angles - Two angles are supplementary angles if the sum of their measures is 180⁰. Adjacent angles - Adjacent angles are two angles that share a common vertex and side, but have no common interior points. Linear pair - Two adjacent angles are a linear pair if their noncommon sides are opposite rays. Vertical angles - Two angles are vertical angles if their sides form two pairs of opposite rays. 1 and 2 are a linear pair. 3 and 6 are vertical angles. 4 and 5 are vertical angles. EXAMPLE 1 Identify complements and supplements In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. EXAMPLE 2 Find measures of a complement and a supplement a. Given that 1 is a complement of 2 and m 2 = 57⁰, find m 1. b. Given that 3 is a supplement of 4 and m 4 = 41⁰, find m 3.

19 EXAMPLE 3 Find angle measures SPORTS The basketball pole forms a pair of supplementary angles with the ground. Find m BCA and m DCA. (1.5 cont.) EXAMPLE 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. EXAMPLE 5 Find angle measures in a linear pair ALGEBRA Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle. CONCEPT SUMMARY Interpreting a Diagram There are some things you can conclude from a diagram, and some you cannot. For example, here are some things that you can conclude from the diagram at the right: All points shown are coplanar. Points A, B, and C are collinear, and B is between A and C. AC, BD, and BE intersect at point B. DBE and EBC are adjacent angles, and ABC is a straight angle. Point E lies in the interior of DBC. In the diagram above, you cannot conclude that AB BC, that DBE EBC, or that ABD is a right angle. This information must be indicated by marks, as shown at the right.

20 1.5 Cont.

21 2.7 Prove Angle Pair Relationships Obj.: Use properties of special pairs of angles. Key Vocabulary Complementary angles - Two angles whose measures have the sum 90. Supplementary angles - Two angles whose measures have the sum 180. Linear pair - Two adjacent angles whose noncommon sides are opposite rays. Vertical angles Two angles whose sides form two pairs of opposite rays. Right Angles Congruence Theorem All right angles are congruent. Abbreviation All Rt. s Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If 1 and 2 are supplementary and 3 and 2 are supplementary, then 1 3. Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If 4 and 5 are complementary and 6 and 5 are complementary, then 4 6. Supp. Th Comp. Th Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 and 2 form a linear pair, so 1 and 2 are supplementary and m 1 +m 2 = 180⁰. Vertical Angles Congruence Theorem Vertical angles are congruent. 1 3, 2 4 EXAMPLE 1 Use right angle congruence Write a proof. GIVEN: JK KL, ML KL PROVE: K L

22 EXAMPLE 2 Use Congruent Supplements Theorem Write a proof. GIVEN: 1 and 2 are supplements. 1 and 4 are supplements. m 2 = 45⁰ PROVE: m 4 = 45⁰ (2.7 cont.) EXAMPLE 3 Use the Vertical Angles Congruence Theorem Prove vertical angles are congruent. GIVEN: 4 is a right angle. PROVE: 2 and 4 are supplementary. EXAMPLE 4 Find angle measures Write and solve an equation to find x. Use x to find the m FKG.

23 2.7 Cont.

24 1.6 Classify Polygons Obj.: Classify polygons. Key Vocabulary Polygon - In geometry, a figure that lies in a plane is called a plane figure. A polygon is a closed plane figure with the following properties: Side - 1. It is formed by three or more line segments called sides. 2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear. Vertex - Each endpoint of a side is a vertex of the polygon. The plural of vertex is vertices. Convex - A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Concave - A polygon that is not convex is called nonconvex or concave. n-gon - The term n-gon, where n is the number of a polygon s sides, can also be used to name a polygon. For example, a polygon with 14 sides is a 14-gon. Equilateral - In an equilateral polygon, all sides are congruent. Equiangular - In an equiangular polygon, all angles in the interior of the polygon are congruent. Regular - A regular polygon is a convex polygon that is both equilateral and equiangular. regular pentagon A polygon can be named by listing the vertices in consecutive order. For example, ABCDE and CDEAB are both correct names for the polygon at the right. cave in convex polygon concave polygon CLASSIFYING POLYGONS A polygon is named by the number of its sides. # of sides Type of polygon # of sides Type of polygon 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n n-gon

25 EXAMPLE 1 Identify polygons Tell whether the figure is a polygon and whether it is convex or concave. (1.6 cont.) EXAMPLE 2 Classify polygons Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. EXAMPLE 3 Find side lengths + ALGEBRA A head of a bolt is shaped like a regular hexagon. The expressions shown represent side lengths of the hexagonal bolt. Find the length of a side.

26 1.6 Cont. (Write these on your paper)

27 2.1 Use Inductive Reasoning Obj.: Describe patterns and use inductive reasoning. Key Vocabulary Conjecture - A conjecture is an unproven statement that is based on observations. Inductive reasoning - You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. Counterexample - A counterexample is a specific case for which the conjecture is false. You can show that a conjecture is false, however, by simply finding one counterexample. EXAMPLE 1 Describe a visual pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. EXAMPLE 2 Describe a number pattern Describe the pattern in the numbers 1, 4, 16, 64,... and write the next three numbers in the pattern. EXAMPLE 3 Make a conjecture Given five noncollinear points, make a conjecture about the number of ways to connect different pairs of the points.

28 EXAMPLE 4 Make and test a conjecture Numbers such as 1, 3, and 5 are called consecutive odd numbers. Make and test a conjecture about the sum of any three consecutive odd numbers. (2.1 cont.) EXAMPLE 5 Find a counterexample A student makes the following conjecture about the difference of two numbers. Find a counterexample to disprove the student s conjecture. Conjecture The difference of any two numbers is always smaller than the larger number. EXAMPLE 6 Real world application The scatter plot shows the average salary of players in the National Football League (NFL) since Make a conjecture based on the graph.

29 2.1 Cont.

30 2.3 Apply Deductive Reasoning Obj.: Use deductive reasoning to form a logical argument. Key Vocabulary Deductive reasoning - Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. **** This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture.**** Laws of Logic KEY CONCEPT Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Law of Syllogism If hypothesis p, then conclusion q. If these statements are true, If hypothesis q, then conclusion r. If hypothesis p, then conclusion r. then this statement is true. EXAMPLE 1 Use the Law of Detachment Use the Law of Detachment to make a valid conclusion in the true situation. a. If two angles have the same measure, then they are congruent. You know that m A = m B. b. Jesse goes to the gym every weekday. Today is Monday. EXAMPLE 2 Use the Law of Syllogism (2.3 cont.) If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. a. If Ron eats lunch today, then he will eat a sandwich. If Ron eats a sandwich, then he will drink a glass of milk.

31 b. If x² > 36, then x² > 30. If x > 6, then x² > 36. c. If a triangle is equilateral, then all of its sides are congruent. If a triangle is equilateral, then all angles in the interior of the triangle are congruent. EXAMPLE 3 Use inductive and deductive reasoning What conclusion can you make about the sum of an odd integer and an odd integer? EXAMPLE 4 Reasoning from a graph Tell whether the statement is the result of inductive reasoning or deductive reasoning. Explain your choice. a. The runner s average speed decreases as time spent running increases. b. The runner s average speed is slower when running for 40 minutes than when running for 10 minutes.

32 2.3 Cont.

33 2.2 Analyze Conditional Statements Obj.: Write definitions as conditional statements. Key Vocabulary Conditional statement - A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion. Converse - To write the converse of a conditional statement, exchange the hypothesis and conclusion. Inverse - To write the inverse of a conditional statement, negate (not) both the hypothesis and the conclusion. Contrapositive - To write the contrapositive, first write the converse and then negate both the hypothesis and the conclusion. If-then form, Hypothesis, Conclusion - When a conditional statement is written in if-then form, the if part contains the hypothesis and the then part contains the conclusion. Here is an example: hypothesis conclusion Negation - The negation of a statement is the opposite of the original statement. Equivalent statements - When two statements are both true or both false, they are called equivalent statements. Perpendicular lines - If two lines intersect to form a right angle, then they are perpendicular lines. You can write line l is perpendicular to line m as l m. KEY CONCEPT l m. Biconditional statement A biconditional statement is a statement that contains the converse and the phrase if and only if. EXAMPLE 1 Rewrite a statement in if-then form Rewrite the conditional statement in if-then form. All vertebrates have a backbone.

34 EXAMPLE 2 Write four related conditional statements Write the if-then form, the converse, the inverse, and the contrapositive of the conditional statement Olympians are athletes. Decide whether each statement is true or false. If-then form (2.2 cont.) Converse Inverse Contrapositive EXAMPLE 3 Use definitions Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. a. AC BD b. AED and BEC are a linear pair. EXAMPLE 4 Write a biconditional Write the definition of parallel lines as a biconditional. Definition: If two lines lie in the same plane and do not intersect, then they are parallel.

35 2.2 Cont.

### Geometry 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

### Chapter 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.

### Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion

### Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

### Geometry 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

### Final 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

### Reasoning and Proof Review Questions

www.ck12.org 1 Reasoning and Proof Review Questions Inductive Reasoning from Patterns 1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...? (a) 81 (b) 64 (c) 121 (d) 56 2. What is the next

### GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

### Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,

### Intermediate 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,

### GEOMETRY 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

### 5.1 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

### GEOMETRY. 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

### Definitions, 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

### Geometry 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

### 2.1. 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

### 11.3 Curves, Polygons and Symmetry

11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon

### This 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: (1-1) Points, Lines, & Planes Topic: (1-2) Segment Measure Quiz

### 1. 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

### Geometry 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

### Show all work for credit. Attach paper as needed to keep work neat & organized.

Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order

### Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs

### DEFINITIONS. 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

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Algebraic Properties and Proofs

Algebraic Properties and Proofs Name You have solved algebraic equations for a couple years now, but now it is time to justify the steps you have practiced and now take without thinking and acting without

### /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

### Mathematics Geometry Unit 1 (SAMPLE)

Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

### Selected 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

### Incenter 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

### Geometry 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

### 56 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

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### 2. 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

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

### Angles 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,

### 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.

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

Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### GEOMETRY - QUARTER 1 BENCHMARK

Name: Class: _ Date: _ GEOMETRY - QUARTER 1 BENCHMARK Multiple Choice Identify the choice that best completes the statement or answers the question. Refer to Figure 1. Figure 1 1. What is another name

### The 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

### Name Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion

Section. Lines That Intersect Circles Lines and Segments That Intersect Circles A chord is a segment whose endpoints lie on a circle. A secant is a line that intersects a circle at two points. A tangent

### Conjunction 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

### Hon Geometry Midterm Review

Class: Date: Hon Geometry Midterm Review Multiple Choice Identify the choice that best completes the statement or answers the question. Refer to Figure 1. Figure 1 1. Name the plane containing lines m

### A 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 mid-point and segment bisector M If a line intersects another line segment

### Lesson 1.1 Building Blocks of Geometry

Lesson 1.1 Building Blocks of Geometry For Exercises 1 7, complete each statement. S 3 cm. 1. The midpoint of Q is. N S Q 2. NQ. 3. nother name for NS is. 4. S is the of SQ. 5. is the midpoint of. 6. NS.

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### Chapter 6 Notes: Circles

Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

### Geometry 8-1 Angles of Polygons

. Sum of Measures of Interior ngles Geometry 8-1 ngles of Polygons 1. Interior angles - The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.

### POTENTIAL 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

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Parallel 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

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

### 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.

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

### 37 Basic Geometric Shapes and Figures

37 Basic Geometric Shapes and Figures In this section we discuss basic geometric shapes and figures such as points, lines, line segments, planes, angles, triangles, and quadrilaterals. The three pillars

### 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.

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

### CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.

### 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.

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

### Chapter 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

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### TIgeometry.com. Geometry. Angle Bisectors in a Triangle

Angle Bisectors in a Triangle ID: 8892 Time required 40 minutes Topic: Triangles and Their Centers Use inductive reasoning to postulate a relationship between an angle bisector and the arms of the angle.

### CHAPTER 6 LINES AND ANGLES. 6.1 Introduction

CHAPTER 6 LINES AND ANGLES 6.1 Introduction In Chapter 5, you have studied that a minimum of two points are required to draw a line. You have also studied some axioms and, with the help of these axioms,

### Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### 3.1 Triangles, Congruence Relations, SAS Hypothesis

Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper

### Unit 2 - Triangles. Equilateral Triangles

Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### Conjectures 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:

### Cumulative Test. 161 Holt Geometry. Name Date Class

Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2

### Integrated Math Concepts Module 10. Properties of Polygons. Second Edition. Integrated Math Concepts. Solve Problems. Organize. Analyze. Model.

Solve Problems Analyze Organize Reason Integrated Math Concepts Model Measure Compute Communicate Integrated Math Concepts Module 1 Properties of Polygons Second Edition National PASS Center 26 National

### Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

### Geometry EOC Practice Test #3

Class: Date: Geometry EOC Practice Test #3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which regular polyhedron has 12 petagonal faces? a. dodecahedron

### absolute value function A function containing the absolute function of a variable.

NYS Mathematics Glossary* Integrated Algebra (*This glossary has been amended from the full SED Commencement Level Glossary of Mathematical Terms (available at http://www.emsc.nysed.gov/ciai/mst/math/glossary/home.html)

### Most popular response to

Class #33 Most popular response to What did the students want to prove? The angle bisectors of a square meet at a point. A square is a convex quadrilateral in which all sides are congruent and all angles

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Geometry Chapter 2 Study Guide

Geometry Chapter 2 Study Guide Short Answer ( 2 Points Each) 1. (1 point) Name the Property of Equality that justifies the statement: If g = h, then. 2. (1 point) Name the Property of Congruence that justifies

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### GPS GEOMETRY Study Guide

GPS GEOMETRY Study Guide Georgia End-Of-Course Tests TABLE OF CONTENTS INTRODUCTION...5 HOW TO USE THE STUDY GUIDE...6 OVERVIEW OF THE EOCT...8 PREPARING FOR THE EOCT...9 Study Skills...9 Time Management...10

Grade FCAT 2.0 Mathematics Sample Answers This booklet contains the answers to the FCAT 2.0 Mathematics sample questions, as well as explanations for the answers. It also gives the Next Generation Sunshine

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

### Geometry EOC Practice Test #2

Class: Date: Geometry EOC Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Rebecca is loading medical supply boxes into a crate. Each supply

### Geometry First Semester Final Exam Review

Name: Class: Date: ID: A Geometry First Semester Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find m 1 in the figure below. PQ parallel.

### The 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

### Set 4: Special Congruent Triangles Instruction

Instruction Goal: To provide opportunities for students to develop concepts and skills related to proving right, isosceles, and equilateral triangles congruent using real-world problems Common Core Standards

### Lesson 18: Looking More Carefully at Parallel Lines

Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using

### Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will

Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles

### Equation 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

### MATH 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

### Geometry Review Flash Cards

point is like a star in the night sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space. point may be represented by a dot on

### The 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

### A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles

### 15. Appendix 1: List of Definitions

page 321 15. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things an undefined object of one type,

### Three-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures

SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### @12 @1. G5 definition s. G1 Little devils. G3 false proofs. G2 sketches. G1 Little devils. G3 definition s. G5 examples and counters

Class #31 @12 @1 G1 Little devils G2 False proofs G3 definition s G4 sketches G5 examples and counters G1 Little devils G2 sketches G3 false proofs G4 examples and counters G5 definition s Jacob Amanda