# Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

Save this PDF as:

Size: px
Start display at page:

Download "Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition."

## Transcription

1 Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and ). Also,, and. By substitution,. Therefore, must be larger than each individual angle. By the Exterior Angle Inequality Theorem, 2. measures greater than m 7 By the Exterior Angle Inequality Theorem, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Similarly, the exterior angle ( 9) is larger than either remote interior angle ( 7 and 6). Therefore, 5 and 9 are both larger than. 3. measures greater than m 2 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is larger than either remote interior angle ( 1 and 2). Therefore, 4 is larger than. 4. measures less than m 9 By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and ). Also,, and (. By substitution,. Therefore, must be larger than each individual angle so is larger than both and. esolutions Manual - Powered by Cognero Page 1

2 List the angles and sides of each triangle in order from smallest to largest. 5. Here it is given that,. Therefore, from angle-side relationships in a triangle, we know that Angle: Side: 6. By the Triangle Sum Theorem, So, From angle-side relationships in a triangle, we know Angle: J, K, L Side: 7. HANG GLIDING The supports on a hang glider form triangles like the one shown. Which is longer the support represented by or the support represented by? Explain your reasoning. Since the angle across from segment is larger than the angle across from, we can conclude that is longer than due to Theorem 5.9, which states that if one side of a triangle is longer than another, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. esolutions Manual - Powered by Cognero Page 2

3 Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 8. measures greater than m 2 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is larger than either remote interior angle ( 1 and 2). Therefore, m 4 > m measures less than m 4 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is greater than either remote interior angle ( 1 and 2).Therefore, m 1< m 4 and m 2< m measures less than m 5 By the Exterior Angle Inequality Theorem, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 7 < m 5 and m 8 < m measures less than m 9 By the Exterior Angle Inequality Theorem, the exterior angle ( 9) is larger than either remote interior angle from two different triangles ( 6 and 7 from one triangle and and from another. Therefore,,,, and. 12. measures greater than m 8 By the Exterior Angle Inequality Theorem, the exterior angle ( 2) is larger than either remote interior angle ( 6 and 8). Similarly, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 2 > m 8 and m 5 > m measures greater than m 7 By the Exterior Angle Inequality Theorem, the exterior angle ( 9) is larger than either remote interior angle ( 6 and 7). Similarly, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 9 > m 7 and m 5 > m 7 esolutions Manual - Powered by Cognero Page 3

4 List the angles and sides of each triangle in order from smallest to largest. 14. The hypotenuse of the right triangle must be greater than the other two sides. Therefore, By Theorem 5.9, the measure of the angle opposite the longer side has a greater measure tahn the angle opposite the shorter side, therefore Angle: Side: 15. Based on the diagram, we see that By Theorem 5.9, the measure of the angle opposite the longer side has a greater measure than the angle opposite the shorter side, therefore Angle: Side: 16. Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: esolutions Manual - Powered by Cognero Page 4

5 17.. Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: 18. By the Triangle Sum Theorem, Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: 19. By the Triangle Sum Theorem, So, Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: esolutions Manual - Powered by Cognero Page 5

6 20. SPORTS Ben, Gilberto, and Hannah are playing Ultimate. Hannah is trying to decide if she should pass to Ben or Gilberto. Which player should she choose in order to have the shorter passing distance? Explain your reasoning. Based on the Triangle Sum Theorem, the measure of the missing angle next to Ben is degrees. Since 48 < 70, the side connecting Hannah to Ben is the shortest, based on Theorem Therefore, Hannah should pass to Ben. 21. RAMPS The wedge below represents a bike ramp. Which is longer, the length of the ramp or the length of the top surface of the ramp? Explain your reasoning using Theorem 5.9. If m X = 90, then, based on the Triangle Sum Theorem, m Y + m Z = 90, so m Y < 90 by the definition of inequality. So m X > m Y. According to Theorem 5.10, if m X > m Y, then the length of the side opposite X must be greater than the length of the side opposite Y. Since is opposite X, and is opposite Y, then YZ > XZ. Therefore YZ, the length of the top surface of the ramp, must be greater than the length of the ramp. esolutions Manual - Powered by Cognero Page 6

7 List the angles and sides of each triangle in order from smallest to largest. 22. Using the Triangle Sum Theorem, we can solve for x, as shown below. degrees and the degrees. Therefore,. By Theorem 5.10, we know that the lengths of sides across from larger angles are longer than those across from shorter angles so. Angle: Side: 23. Using the Triangle Sum Theorem, we can solve for x, as shown below. degrees, degrees and the degrees. Therefore,. By Theorem 5.10, we know that the lengths of sides across from larger angles are longer than those across from shorter angles so. Angle: P, Q, M Side: esolutions Manual - Powered by Cognero Page 7

8 Use the figure to determine which angle has the greatest measure , 5, 6 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 1 is greater than m 5 and m , 4, 6 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 2 is greater than m 4 and m , 4, 5 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 7 is greater than m 4 and m , 11, 12 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 3 is greater than m 11 and m , 9, 14 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and the sum of and are its remote interior angles, the m 3 is greater than m 9 and the sum of the m 11 and m 14. If is greater than the sum of and, then it is greater than the measure of the individual angles. Therefore, the m 3 is greater than m 9 and m 14. esolutions Manual - Powered by Cognero Page 8

9 29. 8, 10, 11 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and the sum of and are its remote interior angles, the is greater than m 10 and the sum of the m 11 and m 14. If is greater than the sum of and, then it is greater than the measure of the individual angles. Therefore, the m 8 is greater than m 10 and m 11. Use the figure to determine the relationship between the measures of the given angles. 30. ABD, BDA The side opposite is, which is of length 13. The side opposite is, which is of length 3. Since by Theorem BCF, CFB The side opposite is, which is of length 15. The side opposite is, which is of length 14. Since by Theorem BFD, BDF The side opposite is, which is of length 12. The side opposite is, which is of length 15. Since by Theorem DBF, BFD The side opposite is, which is of length 5. The side opposite is, which is of length 12. Since by Theorem 5.9. esolutions Manual - Powered by Cognero Page 9

10 Use the figure to determine the relationship between the lengths of the given sides. 34. SM, MR Since are a linear pair, The side opposite is. The side opposite is. In,, since 60 < 70. Therefore, by Theorem 5.10,. 35. RP, MP Since form a straight angle,. The side opposite is. The side opposite is. In,, since 70 < 35. Therefore, by Theorem 5.10,. 36. RQ, PQ The side opposite is. The side opposite is. In,, since 30 < 85. Therefore, by Theorem 5.10, 37. RM, RQ By the Triangle Sum Theorem,. The side opposite PQR is. The side opposite RPQ is. In, m PQR > m RPQ, since 65 > 30. Therefore, by Theorem 5.10, PR > RQ. Also, form a straight angle, and, by the Triangle Sum Theorem,. The side opposite MPR is. The side opposite PMR is. In, m MPR > m PMR, since 75 > 70. Therefore, by Theorem 5.10,. If and, then by the transitive property of inequality, Thus, esolutions Manual - Powered by Cognero Page 10

11 38. HIKING Justin and his family are hiking around a lake as shown in the diagram. Order the angles of the triangle formed by their path from largest to smallest. The side opposite is of length 0.5. The side opposite is of length 0.4. The side opposite is of length Since 0.5 > 0.45 > 0.4, by Theorem 5.9, m 3 > m 1 > m 2. COORDINATE GEOMETRY List the angles of each triangle with the given vertices in order from smallest to largest. Justify your answer. 39. A( 4, 6), B( 2, 1), C(5, 6) Use the Distance Formula to find the side lengths. So, Then, esolutions Manual - Powered by Cognero Page 11

12 40. X( 3, 2), Y(3, 2), Z( 3, 6) Use the Distance Formula to find the side lengths. So, Then, 41. List the side lengths of the triangles in the figure from shortest to longest. Explain your reasoning. AB, BC, AC, CD, BD; In, AB < BC < AC and in, BC < CD < BD. By the figure AC < CD, so BC < AC < CD. 42. MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between the sides of a triangle. a. GEOMETRIC Draw three triangles, including one acute, one obtuse, and one right angle. Label the vertices of each triangle A, B, and C. b. TABULAR Measure the length of each side of the three triangles. Then copy and complete the table. c. TABULAR Create two additional tables like the one above, finding the sum of BC and CA in one table and the sum of AB and CA in the other. esolutions Manual - Powered by Cognero Page 12

13 d. ALGEBRAIC Write an inequality for each of the tables you created relating the measure of the sum of two of the sides to the measure of the third side of a triangle. e. VERBAL Make a conjecture about the relationship between the measure of the sum of two sides of a triangle and the measure of the third side. a. b. It might be easier to use centimeters to measure the sides of your triangles and round to the nearest tenth place, as shown in the table below. Sample answer: c. It might be easier to use centimeters to measure the sides of your triangles and round to the nearest tenth place, as shown in the table below. Sample answer: esolutions Manual - Powered by Cognero Page 13

14 d. Compare the last two columns in each table and write an inequality comparing each of them. AB + BC > CA, BC + CA > AB, AB + CA > BC e. Sample answer: The sum of the measures of two sides of a triangle is always greater than the measure of the third side of the triangle. 43. WRITING IN MATH Analyze the information given in the diagram and explain why the markings must be incorrect. Sample answer: R is an exterior angle to, so, by the Exterior Angle Inequality, m R must be greater than m Q, one of its corresponding remote interior angles. The markings indicate that, indicating that m R = m Q. This is a contradiction of the Exterior Angle Inequality Theorem since can't be both equal to and greater than at the same time. Therefore, the markings are incorrect. esolutions Manual - Powered by Cognero Page 14

15 44. CHALLENGE Using only a ruler, draw such that m A > m B > m C. Justify your drawing. Due to the given information, that, we know that this is a triangle with three different angle measures. Therefore, it also has three different side lengths. You might want to start your construction with the longest side, which would be across from the largest angle measure. Since A is the largest angle, the side opposite it,, is the longest side. Then, make a short side across from the smallest angle measure. Since C is the smallest angle, is the shortest side. Connect points A to C and verify that the length of is between the other two sides. 45. OPEN ENDED Give a possible measure for in shown. Explain your reasoning. Since m C > m B, we know, according to Theorem 5.10, that the side opposite must be greater than the side opposite. Therefore, if AB > AC, then AB > 6 so you need to choose a length for AB that is greater than 6. Sample answer: 10; m C > m B so AB > AC, Therefore, Theorem 5.10 is satisfied since 10 > REASONING Is the base of an isosceles triangle sometimes, always, or never the longest side of the triangle? Explain. To reason through this answer, see if you can sketch isosceles triangles that have a base shorter than the two congruent legs, as well as longer than the two congruent legs. Sometimes; Sample answer: If the measures of the base angles are less than 60 degrees, then the base will be the longest leg. If the measures of the base angles are greater than 60 degrees, then the base will be the shortest leg. esolutions Manual - Powered by Cognero Page 15

16 47. CHALLENGE Use the side lengths in the figure to list the numbered angles in order from smallest to largest given that m 2 = m 5. Explain your reasoning. Walk through these angles one side at a time. Given: m 2 = m 5 The side opposite 5 is the smallest side in that triangle and m 2 = m 5, so we know that m 4 and m 6 are both greater than m 2 and m 5. Also, the side opposite m 6 is greater than the side opposite m 4. So far, we have (m 2 = m 5) < m 4 < m 6. Since the side opposite 2 is greater than the side opposite 1, we know that m 1 is less than m 2 and m 5. Now, we have m 1 < (m 2 = m 5) < m 4 < m 6. From the triangles, m 1 + m 2 + m 3 = 180 and m 4 + m 5 + m 6 = 180. Since m 2 = m 5, m 1 + m 3 must equal m 4 + m 6. Since m 1 is less than m 4, we know that m 3 is must be greater than m 6. The side lengths list in order from smallest to largest is :m 1, m 2 = m 5, m 4, m 6, m WRITING IN MATH Explain why the hypotenuse of a right triangle is always the longest side of the triangle. Based on the Triangle Sum Theorem, we know that the sum of all the angles of a triangle add up to 180 degrees. If one of the angles is 90 degrees, then the sum of the other two angles must equal degrees. If two angles add up to 90 degrees, and neither is 0 degrees, then they each must equal less than 90 degrees, making the sides across from these two acute angles shorter than the side across from the 90 degree angle, according to Theorem Because the hypotenuse is across from the 90 degree angle, the largest angle, then it must be the longest side of the triangle. esolutions Manual - Powered by Cognero Page 16

17 49. STATISTICS The chart shows the number and types of DVDs sold at three stores. According to the information in the chart, which of these statements is true? A The mean number of DVDs sold per store was 56. B Store 1 sold twice as many action and horror films as store 3 sold of science fiction. C Store 2 sold fewer comedy and science fiction than store 3 sold. D The mean number of science fiction DVDs sold per store was 46. Mean of science fiction DVDs is or 46. The correct choice is D. 50. Two angles of a triangle have measures and. What type of triangle is it? F obtuse scalene G obtuse isosceles H acute scalene J acute isosceles By Triangle Sum Theorem, the measure of the third angle is or Since no angle measures are equal, it is a scalene triangle. And one of the angles is obtuse; so, it is an Obtuse Scalene Triangle. The correct choice is F. esolutions Manual - Powered by Cognero Page 17

18 51. EXTENDED RESPONSE At a five-star restaurant, a waiter earns a total of t dollars for working h hours in which he receives \$198 in tips and makes \$2.50 per hour. a. Write an equation to represent the total amount of money the waiter earns. b. If the waiter earned a total of \$213, how many hours did he work? c. If the waiter earned \$150 in tips and worked for 12 hours, what is the total amount of money he earned? a. The equation that represents the total amount of money the waiter earns is: b. Substitute t = 213 in the equation and find h. So, the waiter worked for 6 hours. c. The total amount he earned is \$ SAT/ACT Which expression has the least value? A B C D E 99 = 99 has the least value. So, the correct choice is E. esolutions Manual - Powered by Cognero Page 18

19 In, P is the centroid, KP = 3, and XJ = 8. Find each length. 53. XK Here,. By Centroid Theorem, 54. YJ Since J is the midpoint of XY, esolutions Manual - Powered by Cognero Page 19

20 COORDINATE GEOMETRY Write an equation in slope-intercept form for the perpendicular bisector of the segment with the given endpoints. Justify your answer. 55. D( 2, 4) and E(3, 5) The slope of the segment DE is or So, the slope of the perpendicular bisector is 5. The perpendicular bisector passes through the mid point of the segment DE. The midpoint of DE is or The slope-intercept form for the equation of the perpendicular bisector of DE is: esolutions Manual - Powered by Cognero Page 20

21 56. D( 2, 4) and E(2, 1) The slope of the segment DE is or So, the slope of the perpendicular bisector is The perpendicular bisector passes through the mid point of the segment DE. The midpoint of DE is or The slope-intercept form for the equation of the perpendicular bisector of DE is: esolutions Manual - Powered by Cognero Page 21

22 57. JETS The United States Navy Flight Demonstration Squadron, the Blue Angels, flies in a formation that can be viewed as two triangles with a common side. Write a two-column proof to prove that if T is the midpoint of and. The given information in this proof can help make congruent corresponding sides for the two triangles. If T is the midpoint of, then it would form two congruent segments, and. Using this, along with the other given and the Reflexive Property for the third pair of sides, you can prove these triangles are congruent using SSS. Given: T is the midpoint of Prove: Proof: Statements(Reasons) 1. T is the midpoint of (Given) 2. (Definition of midpoint) 3. (Given) 4. (Reflexive Property) 5. (SSS) 58. POOLS A rectangular pool is 20 feet by 30 feet. The depth of the pool is 60 inches, but the depth of the water is of the depth of the pool. Find each measure to the nearest tenth. (Lesson 1-7) a. the surface area of the pool b. the volume of water in the pool 12 inches = 1 foot. Therefore, 60 inches = 5 feet. a. Surface area of the pool S is: b. The depth of the water is or 3.75ft. The volume of the water in the pool is: esolutions Manual - Powered by Cognero Page 22

23 Determine whether each equation is true or false if x = 8, y = 2, and z = The equation is false for these values The equation is false for these values. The equation is true for these values. esolutions Manual - Powered by Cognero Page 23

### 8-2 The Pythagorean Theorem and Its Converse. Find x.

1 8- The Pythagorean Theorem and Its Converse Find x. 1. hypotenuse is 13 and the lengths of the legs are 5 and x.. equaltothesquareofthelengthofthehypotenuse. The length of the hypotenuse is x and the

### 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

### The Triangle and its Properties

THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three

### 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

### Name Period 10/22 11/1 10/31 11/1. Chapter 4 Section 1 and 2: Classifying Triangles and Interior and Exterior Angle Theorem

Name Period 10/22 11/1 Vocabulary Terms: Acute Triangle Right Triangle Obtuse Triangle Scalene Isosceles Equilateral Equiangular Interior Angle Exterior Angle 10/22 Classify and Triangle Angle Theorems

### 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

### 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

### 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

### 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

### 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

### 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

### Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter

Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,

### 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

### 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,

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### 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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### 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

### 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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### 10.2 45-45 -90 Triangles

Page of 6 0. --0 Triangles Goal Find the side lengths of --0 triangles. Key Words --0 triangle isosceles triangle p. 7 leg of a right triangle p. hypotenuse p. Geo-Activity Eploring an Isosceles Right

### 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.

### 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,

### 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

### http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

### 1-3 Distance and Midpoints. Use the number line to find each measure.

and Midpoints The distance between W and Zis9So,WZ = 9 Use the number line to find each measure 1XY SOLUTION: TIME CAPSULE Graduating classes have buried time capsules on the campus of East Side High School

### 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

### Visualizing Triangle Centers Using Geogebra

Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will

### 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

### 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

### 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:

### Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles

Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use

### 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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 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, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.

Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where

### 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

### Lesson 1: Introducing Circles

IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed

### 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

### POTENTIAL REASONS: Definition of Congruence: Definition of Midpoint: Definition of Angle Bisector:

Sec 1.6 CC Geometry Triangle Proofs Name: POTENTIAL REASONS: Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Geometry: Classifying, Identifying, and Constructing Triangles

Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### 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

### Finding the Measure of Segments Examples

Finding the Measure of Segments Examples 1. In geometry, the distance between two points is used to define the measure of a segment. Segments can be defined by using the idea of betweenness. In the figure

### Lesson 18 Pythagorean Triples & Special Right Triangles

Student Name: Date: Contact Person Name: Phone Number: Teas Assessment of Knowledge and Skills Eit Level Math Review Lesson 18 Pythagorean Triples & Special Right Triangles TAKS Objective 6 Demonstrate

### 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

### 8-5 Angles of Elevation and Depression. The length of the base of the ramp is about 27.5 ft.

1.BIKING Lenora wants to build the bike ramp shown. Find the length of the base of the ramp. The length of the base of the ramp is about 27.5 ft. ANSWER: 27.5 ft 2.BASEBALL A fan is seated in the upper

### Section 7.1 Solving Right Triangles

Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Chapter 5.1 and 5.2 Triangles

Chapter 5.1 and 5.2 Triangles Students will classify triangles. Students will define and use the Angle Sum Theorem. A triangle is formed when three non-collinear points are connected by segments. Each

### NAME DATE PERIOD. Study Guide and Intervention

opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-1 M IO tudy Guide and Intervention isectors, Medians, and ltitudes erpendicular isectors and ngle isectors perpendicular bisector

### 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

### Discovering Math: Exploring Geometry Teacher s Guide

Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional

### 8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

### Analytical Geometry (4)

Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line

### 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

### 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

### 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

### Right Triangles 4 A = 144 A = 16 12 5 A = 64

Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right

### Estimating Angle Measures

1 Estimating Angle Measures Compare and estimate angle measures. You will need a protractor. 1. Estimate the size of each angle. a) c) You can estimate the size of an angle by comparing it to an angle

### Pythagorean Theorem: 9. x 2 2

Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2

### 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

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

### 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

### 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.

### Tangent Properties. Line m is a tangent to circle O. Point T is the point of tangency.

CONDENSED LESSON 6.1 Tangent Properties In this lesson you will Review terms associated with circles Discover how a tangent to a circle and the radius to the point of tangency are related Make a conjecture

### Duplicating Segments and Angles

CONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson, you Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using patty

### 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

### 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

### 9 Right Triangle Trigonometry

www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5

### Session 5 Dissections and Proof

Key Terms for This Session Session 5 Dissections and Proof Previously Introduced midline parallelogram quadrilateral rectangle side-angle-side (SAS) congruence square trapezoid vertex New in This Session

### /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

### Lesson 9.1 The Theorem of Pythagoras

Lesson 9.1 The Theorem of Pythagoras Give all answers rounded to the nearest 0.1 unit. 1. a. p. a 75 cm 14 cm p 6 7 cm 8 cm 1 cm 4 6 4. rea 9 in 5. Find the area. 6. Find the coordinates of h and the radius

### 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

### Geometry. Relationships in Triangles. Unit 5. Name:

Geometry Unit 5 Relationships in Triangles Name: 1 Geometry Chapter 5 Relationships in Triangles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK.

### 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

### 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)

### 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,

### 5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors Equidistant Distance and Perpendicular Bisectors Theorem Hypothesis Conclusion Perpendicular Bisector Theorem Converse of the Perp. Bisector Theorem Locus Applying

### 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

### 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

### 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

### Lesson 3.1 Duplicating Segments and Angles

Lesson 3.1 Duplicating Segments and ngles In Exercises 1 3, use the segments and angles below. Q R S 1. Using only a compass and straightedge, duplicate each segment and angle. There is an arc in each

### Triangles. Triangle. a. What are other names for triangle ABC?

Triangles Triangle A triangle is a closed figure in a plane consisting of three segments called sides. Any two sides intersect in exactly one point called a vertex. A triangle is named using the capital

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

### 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?

### 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

### CAIU Geometry - Relationships with Triangles Cifarelli Jordan Shatto

CK-12 FOUNDATION CAIU Geometry - Relationships with Triangles Cifarelli Jordan Shatto CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12

### Chapter 4: Congruent Triangles

Name: Chapter 4: Congruent Triangles Guided Notes Geometry Fall Semester 4.1 Apply Triangle Sum Properties CH. 4 Guided Notes, page 2 Term Definition Example triangle polygon sides vertices Classifying

### 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

### Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook

Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook Objectives Identify congruent figures and corresponding parts of closed plane figures. Prove that

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### 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

### 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

### Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation

Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units,

### Applications of the Pythagorean Theorem

9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,