8.2 Angle Bisectors of Triangles

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "8.2 Angle Bisectors of Triangles"

Transcription

1 Name lass Date 8.2 ngle isectors of Triangles Essential uestion: How can you use angle bisectors to find the point that is equidistant from all the sides of a triangle? Explore Investigating Distance from a oint to a ine Use a ruler, a protractor, and a piece of tracing paper to investigate points on the bisector of an angle. esource ocker Use the ruler to draw a large angle on tracing paper. abel it. Fold the paper so that coincides with. Open the paper. The crease is the bisector of. lot a point on the bisector. Use the ruler to draw several different segments from point to. Measure the lengths of the segments. Then measure the angle each segment makes with. What do you notice about the shortest segment you can draw from point to? Draw the shortest segment you can from point to. Measure its length. How does its length compare with the length of the shortest segment you drew from point to? Houghton Mifflin Harcourt ublishing ompany eflect 1. uppose you choose a point on the bisector of Z and you draw the perpendicular segment from to and the perpendicular segment from to Z. What do you think will be true about these segments? 2. Discussion What do you think is the best way to measure the distance from a point to a line? Why? Module esson 2

2 Explain 1 pplying the ngle isector Theorem and Its onverse The distance from a point to a line is the length of the perpendicular segment from the point to the line. ou will prove the following theorems about angle bisectors and the sides of the angle they bisect in Exercises 16 and 17. ngle isector Theorem If a point is on the bisector an of angle, then it is equidistant from the sides of the angle., so =. onverse of the ngle isector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. =, so Example 1 Find each measure. M 12.8 M M is the bisector of, so M = M = m D, given that m = 112 eflect 74 D 74 ince D = D, D, and D, you know that D bisects by the o, m D = 1 m =. 2 Theorem. Houghton Mifflin Harcourt ublishing ompany 3. In the onverse of the ngle isector Theorem, why is it important to say that the point must be in the interior of the angle? Module esson 2

3 our Turn Find each measure m M, given that m M = M 62 Explain 2 onstructing an Inscribed ircle circle is inscribed in a polygon if each side of the polygon is tangent to the circle. In the figure, circle is inscribed in quadrilateral WZ and this circle is called the incircle (inscribed circle) of the quadrilateral. In order to construct the incircle of a triangle, you need to find the center of the circle. This point is called the incenter of the triangle. Example 2 Use a compass and straightedge to construct the inscribed circle of. W Z tep 1 The center of the inscribed circle must be equidistant from and. What is the set of points equidistant from and? onstruct this set of points. Houghton Mifflin Harcourt ublishing ompany eflect tep 2 The center must also be equidistant from and. What is the set of points equidistant from and? onstruct this set of points. tep 3 The center must lie at the intersection of the two sets of points you constructed. abel this point. tep 4 lace the point of your compass at and open the compass until the pencil just touches a side of. Then draw the inscribed circle. 6. uppose you started by constructing the set of points equidistant from and, and then constructed the set of points equidistant from and. Would you have found the same center point? heck by doing this construction. Module esson 2

4 Explain 3 Using roperties of ngle isectors s you have seen, the angle bisectors of a triangle are concurrent. The point of concurrency is the incenter of the triangle. Incenter Theorem The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Z = = Z Example 3 V and V are angle bisectors of. Find each measure. the distance from V to V is the incenter of. y the Incenter Theorem, V is equidistant from the sides of. The distance from V to is 7.3. o the distance from V to is also W V m V V is the bisector of. m = 2 ( ) = Triangle um Theorem + + m = 180 ubtract from each side. m = V is the bisector of. m V = 1 2 ( ) = eflect 7. In art, is there another distance you can determine? Explain. our Turn and are angle bisectors of. Find each measure. 8. the distance from to 9. m Houghton Mifflin Harcourt ublishing ompany Module esson 2

5 Elaborate 10. and are the circumcenter and incenter of T, but not necessarily in that order. Which point is the circumcenter? Which point is the incenter? Explain how you can tell without constructing any bisectors. T 11. omplete the table by filling in the blanks to make each statement true. ircumcenter Incenter Definition The point of concurrency of the The point of concurrency of the Distance Equidistant from the Equidistant from the ocation (Inside, Outside, On) an be the triangle lways the triangle 12. Essential uestion heck-in How do you know that the intersection of the bisectors of the angles of a triangle is equidistant from the sides of the triangle? Evaluate: Homework and ractice Houghton Mifflin Harcourt ublishing ompany 1. Use a compass and straightedge to investigate points on the bisector of an angle. On a separate piece of paper, draw a large angle. a. onstruct the bisector of. b. hoose a point on the angle bisector you constructed. abel it. onstruct a perpendicular through to each side of. c. Explain how to use a compass to show that is equidistant from the sides of. Find each measure. 2. V V 3. m M, given that m = 63 W 4.9 Online Homework Hints and Help Extra ractice M Module esson 2

6 4. D 5. m HF, given that m GF = 45 D H G 10.2 F onstruct an inscribed circle for each triangle N M F and EF are angle bisectors of DE. Find each measure. 8. the distance from F to D 9. m FED 17 F G D E T and are angle bisectors of T. Find each measure. 10. the distance from to 11. m T 42 T Houghton Mifflin Harcourt ublishing ompany Module esson 2

7 Find each measure V 6y y + 6 D 5m - 3 U 2m + 9 V 14. m 15. m GDF (2x + 1)º (3x - 9)º M (6y + 3) G 2.7 D (7y - 3) H 2.7 F 16. omplete the following proof of the ngle isector Theorem. Given: rove: bisects., = Houghton Mifflin Harcourt ublishing ompany tatements 1. bisects., easons 3. and are right angles. 3. Definition of perpendicular ll right angles are congruent eflexive roperty of ongruence Triangle ongruence Theorem = 8. ongruent segments have the same length. Module esson 2

8 17. omplete the following proof of the onverse of the ngle isector Theorem. Given: V, VZ Z, V = VZ. rove: V bisects Z. V Z tatements 1. V, VZ Z, V = VZ 1. easons 2. V and VZ are right angles V V V ZV V ZV omplete the following proof of the Incenter Theorem. Given:,, and bisect, and, respectively.,, Z rove: = = Z et be the incenter of. ince lies on the bisector of, = by the Theorem. imilarly, also, so = Z. Therefore, = = Z, by the. 19. city plans to build a firefighter s monument in a triangular park between three streets. Draw a sketch on the figure to show where the city should place the monument so that it is the same distance from all three streets. ustify your sketch. Fillmore treet uchanan treet olk treet Z Houghton Mifflin Harcourt ublishing ompany Image redits: ep oig/ lamy Module esson 2

9 20. school plans to place a flagpole on the lawn so that it is equidistant from Mercer treet and Houston treet. They also want the flagpole to be equidistant from a water fountain at W and a bench at. Find the point F where the school should place the flagpole. Mark the point on the figure and explain your answer. Mercer treet W Houston treet 21. is the incenter of. Determine whether each statement is true or false. elect the correct answer for each lettered part. a. oint must lie on the perpendicular bisector of. True False b. oint must lie on the angle bisector of. True False c. If is 23 mm long, then must be 23 mm long. True False d. If the distance from point to is x, then the distance from point to must be x. True False e. The perpendicular segment from point to is longer than the perpendicular segment from point to. True False Houghton Mifflin Harcourt ublishing ompany Module esson 2

10 H.O.T. Focus on Higher Order Thinking 22. What If? In the Explore, you constructed the angle bisector of acute and found that if a point is on the bisector, then it is equidistant from the sides of the angle. Would you get the same results if were a straight angle? Explain. 23. Explain the Error student was asked to draw the incircle for. He constructed angle bisectors as shown. Then he drew a circle through points,, and. Describe the student s error. esson erformance Task Teresa has just purchased a farm with a field shaped like a right triangle. The triangle has the measurements shown in the diagram. Teresa plans to install central pivot irrigation in the field. In this type of irrigation, a circular region of land is irrigated by a long arm of sprinklers the radius of the circle that rotates around a central pivot point like the hands of a clock, dispensing water as it moves. a. Describe how she can find where to locate the pivot. 24 yd 51 yd 45 yd b. Find the area of the irrigation circle. To find the radius, r, of a circle inscribed in a triangle with sides of length a, b, and c, you can use the formula r = k = 1 (a + b + c). 2 k (k - a) (k - b) (k - c) k c. bout how much of the field will not be irrigated?, where Houghton Mifflin Harcourt ublishing ompany Module esson 2

Duplicating Segments and Angles

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

More information

Duplicating Segments and Angles

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

More information

Lesson 3.1 Duplicating Segments and Angles

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

More information

NAME DATE PERIOD. Study Guide and Intervention

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

More information

Warm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example:

Warm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example: Geometr hapter 12 Notes - 1 - Warm Up #23: Review of ircles 1.) central angle of a circle is an angle with its verte at the of the circle. Eample: X 80 2.) n arc is a section of a circle. Eamples:, 3.)

More information

Chapter 6 Notes: Circles

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

More information

Chapters 6 and 7 Notes: Circles, Locus and Concurrence

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

More information

For the circle above, EOB is a central angle. So is DOE. arc. The (degree) measure of ù DE is the measure of DOE.

For the circle above, EOB is a central angle. So is DOE. arc. The (degree) measure of ù DE is the measure of DOE. efinition: circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. We use the symbol to represent a circle. The a line segment from the center

More information

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,

More information

Use Angle Bisectors of Triangles

Use Angle Bisectors of Triangles 5.3 Use ngle isectors of Triangles efore ou used angle bisectors to find angle relationships. ow ou will use angle bisectors to find distance relationships. Why? So you can apply geometry in sports, as

More information

Notes on Perp. Bisectors & Circumcenters - Page 1

Notes on Perp. Bisectors & Circumcenters - Page 1 Notes on Perp. isectors & ircumcenters - Page 1 Name perpendicular bisector of a triangle is a line, ray, or segment that intersects a side of a triangle at a 90 angle and at its midpoint. onsider to the

More information

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

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

More information

Geometry Chapter 5 - Properties and Attributes of Triangles Segments in Triangles

Geometry Chapter 5 - Properties and Attributes of Triangles Segments in Triangles Geometry hapter 5 - roperties and ttributes of Triangles Segments in Triangles Lesson 1: erpendicular and ngle isectors equidistant Triangle congruence theorems can be used to prove theorems about equidistant

More information

Tangents to Circles. Circle The set of all points in a plane that are equidistant from a given point, called the center of the circle

Tangents to Circles. Circle The set of all points in a plane that are equidistant from a given point, called the center of the circle 10.1 Tangents to ircles Goals p Identify segments and lines related to circles. p Use properties of a tangent to a circle. VOULRY ircle The set of all points in a plane that are equidistant from a given

More information

Circle Name: Radius: Diameter: Chord: Secant:

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

More information

CONGRUENCE BASED ON TRIANGLES

CONGRUENCE BASED ON TRIANGLES HTR 174 5 HTR TL O ONTNTS 5-1 Line Segments ssociated with Triangles 5-2 Using ongruent Triangles to rove Line Segments ongruent and ngles ongruent 5-3 Isosceles and quilateral Triangles 5-4 Using Two

More information

Name Geometry Exam Review #1: Constructions and Vocab

Name Geometry Exam Review #1: Constructions and Vocab Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make

More information

Lesson 5-3: Concurrent Lines, Medians and Altitudes

Lesson 5-3: Concurrent Lines, Medians and Altitudes Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special

More information

Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the

Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the ngle Measure Vocabulary degree ray opposite rays angle sides vertex interior exterior right angle acute angle obtuse angle angle bisector tudy ip eading Math Opposite rays are also known as a straight

More information

Int. Geometry Unit 2 Quiz Review (Lessons 1-4) 1

Int. Geometry Unit 2 Quiz Review (Lessons 1-4) 1 Int. Geometry Unit Quiz Review (Lessons -4) Match the examples on the left with each property, definition, postulate, and theorem on the left PROPRTIS:. ddition Property of = a. GH = GH. Subtraction Property

More information

39 Symmetry of Plane Figures

39 Symmetry of Plane Figures 39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

More information

PARALLEL LINES CHAPTER

PARALLEL LINES CHAPTER HPTR 9 HPTR TL OF ONTNTS 9-1 Proving Lines Parallel 9-2 Properties of Parallel Lines 9-3 Parallel Lines in the oordinate Plane 9-4 The Sum of the Measures of the ngles of a Triangle 9-5 Proving Triangles

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

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,

More information

6.2 PLANNING. Chord Properties. Investigation 1 Defining Angles in a Circle

6.2 PLANNING. Chord Properties. Investigation 1 Defining Angles in a Circle LESSN 6.2 You will do foolish things, but do them with enthusiasm. SINIE GRIELL LETTE Step 1 central Step 1 angle has its verte at the center of the circle. Step 2 n Step 2 inscribed angle has its verte

More information

Geometry Unit 10 Notes Circles. Syllabus Objective: 10.1 - The student will differentiate among the terms relating to a circle.

Geometry Unit 10 Notes Circles. Syllabus Objective: 10.1 - The student will differentiate among the terms relating to a circle. Geometry Unit 0 Notes ircles Syllabus Objective: 0. - The student will differentiate among the terms relating to a circle. ircle the set of all points in a plane that are equidistant from a given point,

More information

The Geometry of Piles of Salt Thinking Deeply About Simple Things

The Geometry of Piles of Salt Thinking Deeply About Simple Things The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word

More information

CK-12 Geometry: Parts of Circles and Tangent Lines

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.

More information

GEOMETRY. Constructions OBJECTIVE #: G.CO.12

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

More information

Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24

Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 24 esources: Springoard- Geometry Unit Overview In this unit, students will study formal definitions of basic figures,

More information

Elementary triangle geometry

Elementary triangle geometry Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors

More information

IMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.

IMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao. ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.

More information

1.2 Informal Geometry

1.2 Informal Geometry 1.2 Informal Geometry Mathematical System: (xiomatic System) Undefined terms, concepts: Point, line, plane, space Straightness of a line, flatness of a plane point lies in the interior or the exterior

More information

A geometric construction is a drawing of geometric shapes using a compass and a straightedge.

A geometric construction is a drawing of geometric shapes using a compass and a straightedge. Geometric Construction Notes A geometric construction is a drawing of geometric shapes using a compass and a straightedge. When performing a geometric construction, only a compass (with a pencil) and a

More information

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have

More information

EXAMPLE. Step 1 Draw a ray with endpoint C. Quick Check

EXAMPLE. Step 1 Draw a ray with endpoint C. Quick Check -7. lan -7 asic onstructions Objectives o use a compass and a straightedge to construct congruent segments and congruent angles o use a compass and a straightedge to bisect segments and angles Examples

More information

Geometry SOL G.11 G.12 Circles Study Guide

Geometry SOL G.11 G.12 Circles Study Guide Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter-10-circles-flashcardsflash-cards/).

More information

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. Which type of construction is best represented by the steps given above? 1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

More information

11 th Annual Harvard-MIT Mathematics Tournament

11 th Annual Harvard-MIT Mathematics Tournament 11 th nnual Harvard-MIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1. [] How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit

More information

A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:

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

More information

Incenter and Circumcenter Quiz

Incenter and Circumcenter Quiz Name: lass: ate: I: Incenter and ircumcenter Quiz Multiple hoice Identify the choice that best completes the statement or answers the question.. The diagram below shows the construction of the center of

More information

Power of a Point Solutions

Power of a Point Solutions ower of a oint Solutions Yufei Zhao Trinity ollege, ambridge yufei.zhao@gmail.com pril 2011 ractice problems: 1. Let Γ 1 and Γ 2 be two intersecting circles. Let a common tangent to Γ 1 and Γ 2 touch Γ

More information

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.

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

More information

Mathsercise. Revision Practice for Target C grade GCSE Geometry

Mathsercise. Revision Practice for Target C grade GCSE Geometry Mathsercise Revision Practice for Target grade GSE Geometry Mathsercise- ngles Work out the size of angles and y. Give reasons for your answers 5 o y ngles Work out the size of angles and y. Give reasons

More information

MATH 139 FINAL EXAM REVIEW PROBLEMS

MATH 139 FINAL EXAM REVIEW PROBLEMS MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice

More information

Geometer s Sketchpad. Discovering the incenter of a triangle

Geometer s Sketchpad. Discovering the incenter of a triangle Geometer s Sketchpad Discovering the incenter of a triangle Name: Date: 1.) Open Geometer s Sketchpad (GSP 4.02) by double clicking the icon in the Start menu. The icon looks like this: 2.) Once the program

More information

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures 8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

More information

Chapter 5: Relationships within Triangles

Chapter 5: Relationships within Triangles Name: Chapter 5: Relationships within Triangles Guided Notes Geometry Fall Semester CH. 5 Guided Notes, page 2 5.1 Midsegment Theorem and Coordinate Proof Term Definition Example midsegment of a triangle

More information

Unit 2 - Triangles. Equilateral Triangles

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

More information

Radius, diameter, circumference, π (Pi), central angles, Pythagorean relationship. about CIRCLES

Radius, diameter, circumference, π (Pi), central angles, Pythagorean relationship. about CIRCLES Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 1 Chapter 8 in textbook (p. 384 420) 5/50 or 10% on 2011 CRT: 5 Multiple Choice WHAT YOU SHOULD ALREADY KNOW: Radius, diameter, circumference, π (Pi), central

More information

Geometry Chapter 5 Relationships Within Triangles

Geometry Chapter 5 Relationships Within Triangles Objectives: Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 To use properties of midsegments to solve problems. To use properties of perpendicular bisectors and angle bisectors. To identify

More information

Centers of Triangles Learning Task. Unit 3

Centers of Triangles Learning Task. Unit 3 Centers of Triangles Learning Task Unit 3 Course Mathematics I: Algebra, Geometry, Statistics Overview This task provides a guided discovery and investigation of the points of concurrency in triangles.

More information

7. 6 Justifying Constructions

7. 6 Justifying Constructions 31 7. 6 Justifying Constructions A Solidify Understanding Task CC BY THOR https://flic.kr/p/9qkxv Compass and straightedge constructions can be justified using such tools as: the definitions and properties

More information

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

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,

More information

Lesson 1: Introducing Circles

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

More information

Circle geometry theorems

Circle geometry theorems Circle geometry theorems http://topdrawer.aamt.edu.au/geometric-reasoning/big-ideas/circlegeometry/angle-and-chord-properties Theorem Suggested abbreviation Diagram 1. When two circles intersect, the line

More information

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

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

More information

Geometry Chapter 10 Study Guide Name

Geometry Chapter 10 Study Guide Name eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.

More information

Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013 Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

More information

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

More information

Three Lemmas in Geometry

Three Lemmas in Geometry Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle

More information

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points. Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

More information

12 CONSTRUCTIONS AND LOCI

12 CONSTRUCTIONS AND LOCI 12 ONSTRUTIONS N LOI rchitects make scale drawings of projects they are working on for both planning and presentation purposes. Originally these were done on paper using ink, and copies had to be made

More information

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangent-hord Theorem, Intersecting hord Theorem and Tangent-secant Theorem utline asic definitions and facts on circles The

More information

Finding Angle Measures. Solve. 2.4 in. Label the diagram. Draw AE parallel to BC. Simplify. Use a calculator to find the square root. 14 in.

Finding Angle Measures. Solve. 2.4 in. Label the diagram. Draw AE parallel to BC. Simplify. Use a calculator to find the square root. 14 in. 1-1 1. lan bjectives 1 To use the relationship between a radius and a tangent To use the relationship between two tangents from one point amples 1 inding ngle Measures Real-World onnection inding a Tangent

More information

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1 Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

More information

5.1 Midsegment Theorem and Coordinate Proof

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

More information

EXPECTED BACKGROUND KNOWLEDGE

EXPECTED BACKGROUND KNOWLEDGE MOUL - 3 oncurrent Lines 12 ONURRNT LINS You have already learnt about concurrent lines, in the lesson on lines and angles. You have also studied about triangles and some special lines, i.e., medians,

More information

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

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,

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

CCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:

CCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name: GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships

More information

The Protractor Postulate and the SAS Axiom. Chapter The Axioms of Plane Geometry

The Protractor Postulate and the SAS Axiom. Chapter The Axioms of Plane Geometry The Protractor Postulate and the SAS Axiom Chapter 3.4-3.7 The Axioms of Plane Geometry The Protractor Postulate and Angle Measure The Protractor Postulate (p51) defines the measure of an angle (denoted

More information

Seattle Public Schools KEY to Review Questions for the Washington State Geometry End of Course Exam

Seattle Public Schools KEY to Review Questions for the Washington State Geometry End of Course Exam Seattle Public Schools KEY to Review Questions for the Washington State Geometry End of ourse Exam 1) Which term best defines the type of reasoning used below? bdul broke out in hives the last four times

More information

Conjectures. Chapter 2. Chapter 3

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

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

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

More information

GEOMETRY COMMON CORE STANDARDS

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,

More information

The Four Centers of a Triangle. Points of Concurrency. Concurrency of the Medians. Let's Take a Look at the Diagram... October 25, 2010.

The Four Centers of a Triangle. Points of Concurrency. Concurrency of the Medians. Let's Take a Look at the Diagram... October 25, 2010. Points of Concurrency Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency. Example: x M w y M is the point of

More information

Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

More information

G7-3 Measuring and Drawing Angles and Triangles Pages

G7-3 Measuring and Drawing Angles and Triangles Pages G7-3 Measuring and Drawing Angles and Triangles Pages 102 104 Curriculum Expectations Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m3, 7m4, 7m46 WNCP: 6SS1, review, [T, R, V] Vocabulary angle vertex arms acute

More information

1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always 3. 1 4. 3 5. 1 6. 1 7a. True 7b. True 7c. True 7d. True 7e. True 8.

1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always 3. 1 4. 3 5. 1 6. 1 7a. True 7b. True 7c. True 7d. True 7e. True 8. 1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always 3. 1 4. 3 5. 1 6. 1 7a. True 7b. True 7c. True 7d. True 7e. True 8. 3 and 13 9. a 4, c 26 10. 8 11. 20 12. 130 13 12 14. 10 15.

More information

Geometry in a Nutshell

Geometry in a Nutshell Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

More information

Classifying Quadrilaterals

Classifying Quadrilaterals 1 lassifying Quadrilaterals Identify and sort quadrilaterals. 1. Which of these are parallelograms?,, quadrilateral is a closed shape with 4 straight sides. trapezoid has exactly 1 pair of parallel sides.

More information

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1.

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Name: lass: ate: I: hapter 1 Exam Multiple hoice Identify the choice that best completes the statement or answers the question. 1. bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = c. m = 40 b. m =

More information

Advanced Euclidean Geometry

Advanced Euclidean Geometry dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line

More information

Points of Concurrency Related to Archaeology Grade Ten

Points of Concurrency Related to Archaeology Grade Ten Ohio Standards Connection: Geometry and Spatial Sense Benchmark A Formally define geometric figures. Indicator 1 Formally define and explain key aspects of geometric figures, including: a. interior and

More information

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA- (x₂-x₁)²+(y₂-y₁)² Find the distance between the points ( -3,2) and

More information

5-1 Perpendicular and Angle Bisectors

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

More information

22.1 Interior and Exterior Angles

22.1 Interior and Exterior Angles Name Class Date 22.1 Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Resource Locker Explore 1 Exploring Interior

More information

New York State Student Learning Objective: Regents Geometry

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

More information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

Geo 9 1 Circles 9-1 Basic Terms associated with Circles and Spheres. Radius. Chord. Secant. Diameter. Tangent. Point of Tangency.

Geo 9 1 Circles 9-1 Basic Terms associated with Circles and Spheres. Radius. Chord. Secant. Diameter. Tangent. Point of Tangency. Geo 9 1 ircles 9-1 asic Terms associated with ircles and Spheres ircle Given Point = Given distance = Radius hord Secant iameter Tangent Point of Tangenc Sphere Label ccordingl: ongruent circles or spheres

More information

Final Review Geometry A Fall Semester

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

More information

*1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles.

*1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review

More information

Geometry Made Easy Handbook Common Core Standards Edition

Geometry Made Easy Handbook Common Core Standards Edition Geometry Made Easy Handbook ommon ore Standards Edition y: Mary nn asey. S. Mathematics, M. S. Education 2015 Topical Review ook ompany, Inc. ll rights reserved. P. O. ox 328 Onsted, MI. 49265-0328 This

More information

Conjectures for Geometry for Math 70 By I. L. Tse

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:

More information

EUCLIDEAN GEOMETRY: (±50 marks)

EUCLIDEAN GEOMETRY: (±50 marks) ULIN GMTRY: (±50 marks) Grade theorems:. The line drawn from the centre of a circle perpendicular to a chord bisects the chord. 2. The perpendicular bisector of a chord passes through the centre of the

More information

Let s Talk About Symmedians!

Let s Talk About Symmedians! Let s Talk bout Symmedians! Sammy Luo and osmin Pohoata bstract We will introduce symmedians from scratch and prove an entire collection of interconnected results that characterize them. Symmedians represent

More information

Inscribed Angle Theorem and Its Applications

Inscribed Angle Theorem and Its Applications : Student Outcomes Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Recognize and use different

More information

CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry. Chapter 2 CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

More information

Math 3372-College Geometry

Math 3372-College Geometry Math 3372-College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang Contents

More information

Unit 6 Grade 7 Geometry

Unit 6 Grade 7 Geometry Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson

More information