To use properties of perpendicular bisectors and angle bisectors

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "To use properties of perpendicular bisectors and angle bisectors"

Transcription

1 5-2 erpendicular and ngle isectors ommon ore tate tandards G-O..9 rove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. lso G-..5 1, 3, 4, 5, 8 Objective o use properties of perpendicular bisectors and angle bisectors onfused? ry drawing a diagram to straighten yourself out. HEIL IE You hang a bulletin board over your desk using string. he bulletin board is crooked. When you straighten the bulletin board, what type of triangle does the string form with the top of the board? How do you know? Visualize the vertical line along the wall that passes through the nail. What relationships exist between this line and the top edge of the straightened bulletin board? Explain. In the olve It, you thought about the relationships that must exist in order for a bulletin board to hang straight. You will explore these relationships in this lesson. Lesson L VocabularyV equidistant distance from a point to a line Essential Understanding here is a special relationship between the points on the perpendicular bisector of a segment and the endpoints of the segment. In the diagram below on the left, < > is the perpendicular bisector of. < > is perpendicular to at its midpoint. In the diagram on the right, and are drawn to complete and. You should recognize from your work in hapter 4 that. o you can conclude that, or that =. point is equidistant from two objects if it is the same distance from the objects. o point is equidistant from points and. his suggests a proof of heorem 5-2, the erpendicular isector heorem. Its converse is also true and is stated as heorem hapter 5 elationships Within riangles

2 heorem 5-2 erpendicular isector heorem heorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If... hen... < > # and = = You will prove heorem 5-2 in Exercise 32. heorem 5-3 onverse of the erpendicular isector heorem heorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If... = hen... < > # and = You will prove heorem 5-3 in Exercise 33. roblem 1 Using the erpendicular isector heorem How do you know is the perpendicular bisector of? he markings in the diagram show that is perpendicular to at the midpoint of. lgebra What is the length of? is the perpendicular bisector of, so is equidistant from and. = erpendicular isector heorem 4x = 6x - 10 ubstitute 4x for and 6x - 10 for. - 2x = - 10 ubtract 6x from each side. 4x 6x 10 x = 5 ivide each side by - 2. Now find. = 4x = 4(5) = 20 ubstitute 5 for x. Got It? 1. What is the length of? 3n 1 5n 7 Lesson 5-2 erpendicular and ngle isectors 293

3 roblem 2 How do you find points that are equidistant from two given points? y the onverse of the erpendicular isector heorem, points equidistant from two given points are on the perpendicular bisector of the segment that joins the two points. Using a erpendicular isector park director wants to build a -shirt stand equidistant from the ollin oaster and the paceship hoot. What are the possible locations of the stand? Explain. addle boats paceship hoot ollin oaster erry-go-round o be equidistant from the two rides, the stand should be on the perpendicular bisector of the segment connecting the rides. Find the midpoint of and draw line / through perpendicular to. he possible locations of the stand are all the points on line /. Got It? 2. a. uppose the director wants the -shirt stand to be equidistant from the paddle boats and the paceship hoot. What are the possible locations? b. easoning an you place the -shirt stand so that it is equidistant from hsm11gmse_0502_t13165 the paddle boats, the paceship hoot, and the ollin oaster? Explain. Essential Understanding here is a special relationship between the points on the bisector of an angle and the sides of the angle. he distance from a point to a line is the length of the perpendicular segment from the point to the line. his distance is also the length of the shortest segment from the point to the line. You will prove this in Lesson 5-6. In the figure at the right, the distances from to / and from to / are represented by the red segments. > In the diagram, is the bisector of. If you measure the lengths of the perpendicular segments from to the two sides of hsm11gmse_0502_t06088 the angle, you will find that the lengths are equal. oint is equidistant from the sides of the angle. 294 hapter 5 elationships Within riangles hsm11gmse_0502_t06089

4 heorem 5-4 ngle isector heorem heorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. If... > bisects, # >, and # > hen... = You will prove heorem 5-4 in Exercise 34. heorem 5-5 onverse of the ngle isector heorem heorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. If... # >, # >, and = hen... > bisects You will prove heorem 5-5 in Exercise 35. roblem 3 lgebra What is the length of? Using the ngle isector heorem 2x 25 N > bisects LN. # NL > and # N >. he length of Use the ngle isector heorem to write an equation you can solve for x. L 7x = ngle isector heorem 7x = 2x + 25 ubstitute. N How can you use the expression given for to check your answer? ubstitute 5 for x in the expression 2x + 25 and verify that the result is 35. 5x = 25 ubtract 2x from each side. x = 5 ivide each side by 5. Now find. = 7x = 7(5) = 35 ubstitute 5 for x. Got It? 3. What is the length of F? 6x 3 F 4x 9 Lesson 5-2 erpendicular and ngle isectors 295

5 Lesson heck o you know HOW? Use the figure at the right for Exercises What is the relationship between and? 2. What is the length of? 3. What is the length of? 15 E 18 o you UNEN? HEIL IE 4. Vocabulary raw a line and a point not on the line. raw the segment that represents the distance from the point to the line. 5. Writing oint is in the interior of LOX. escribe how you can determine whether is on the bisector of LOX without drawing the angle bisector. ractice and roblem-olving Exercises HEIL IE ractice Use the figure at the right for Exercises 6 8. ee roblem What is the relationship between and JK? 9x 18 3x 7. What is value of x? 8. Find J. J K eading aps For Exercises 9 and 10, use the map of a part of anhattan. 9. Which school is equidistant from the subway stations at Union quare and 14th treet? How do you know? 10. Is t. Vincent s Hospital equidistant from Village Kids Nursery chool and Legacy chool? How do you know? 11. Writing On a piece of paper, mark a point H for home and a point for school. escribe how to find the set of points equidistant from H and. 14th t 8th ve NY useum chool t. Vincent s Hospital 7th ve Village Kids Nursery chool ee roblem 2. = ubway tation 6th ve roadway oleman chool Xavier chool Legacy chool Union quare Use the figure at the right for Exercises ee roblem ccording to the diagram, how far is L from HK >? From HF >? 13. How is HL > related to KHF? Explain. 14. Find the value of y. 15. Find m KHL and m FHL. K 27 6y H L F (4y 18) 296 hapter 5 elationships Within riangles

6 16. lgebra Find x, JK, and J. 17. lgebra Find y,, and U. x 5 K 5y J L 3y 6 2x 7 V U pply lgebra Use the figure at the right for Exercises Find the value of x. 19. Find W. 20. Find WZ. 2x Y 21. What kind of triangle is WZ? Explain. 22. If is on the perpendicular bisector of Z, then is? from and Z, or? =?. W 3x 5 Z 23. hink bout a lan In the diagram at the right, the soccer goalie will prepare for a shot from the player at point by moving out to a point on XY. o have the best chance of stopping the ball, should the goalie stand at the point on XY that lies on the perpendicular bisector of GL or at the point on XY that lies on the bisector of GL? Explain your reasoning. How can you draw a diagram to help? Would the goalie want to be the same distance from G and L or from G and L? X G Y L 24. a. onstructions raw E. onstruct the angle bisector of the angle. b. easoning Use the converse of the angle bisector theorem to justify your construction. 25. a. onstructions raw. onstruct the perpendicular bisector of to construct. b. easoning Use the perpendicular bisector theorem to justify that your construction is an isosceles triangle. 26. Write heorems 5-2 and 5-3 as a single biconditional statement. 27. Write heorems 5-4 and 5-5 as a single biconditional statement. Lesson 5-2 erpendicular and ngle isectors 297

7 28. Error nalysis o prove that is isosceles, a student began by stating that since is on the segment perpendicular to, is equidistant from the endpoints of. What is the error in the student s reasoning? Writing etermine whether must be on the bisector of jx. Explain. 29. X X X 32. rove the erpendicular roof isector heorem. Given: < > #, < > bisects rove: = 33. rove the onverse of the roof erpendicular isector heorem. Given: = with # at. rove: is on the perpendicular bisector of. 34. rove the ngle roof isector heorem. Given: > bisects, # >, # > rove: = 35. rove the onverse of the roof ngle isector heorem. Given: # >, # >, = rove: > bisects. 36. oordinate Geometry Use points (6, 8), O(0, 0), and (10, 0). a. Write equations of lines /and m such that / # < O > at and m # < O > at. b. Find the intersection of lines / and m. c. how that =. d. Explain why is on the bisector of O. 298 hapter 5 elationships Within riangles

8 hallenge 37.,, and are three noncollinear points. escribe and sketch a line in plane such that points,, and are equidistant from the line. Justify your response. E 38. easoning is the intersection of the perpendicular bisectors of two sides of. Line / is perpendicular to plane at. Explain why a point E on / is equidistant from,, and. (Hint: ee page 48, Exercise 33. Explain why E E E.) tandardized est rep / 39. For (1, 3) and (1, 9), which point lies on the perpendicular bisector of? (3, 3) (1, 5) (6, 6) (3, 12) 40. What is the converse of the following conditional statement? If a triangle is isosceles, then it has two congruent angles. If a triangle is isosceles, then it has two congruent sides. If a triangle has congruent sides, then it is equilateral. If a triangle has two congruent angles, then it is isosceles. If a triangle is not isosceles, then it does not have two congruent angles. 41. Which figure represents the statement bisects? hort esponse 42. he line y = 7 is the perpendicular bisector of the segment with endpoints (2, 10) and (2, k). What is the value of k? Explain your reasoning. ixed eview 43. Find the value of x in the figure at the right and 2 are complementary and 1 and 3 are supplementary. If m 2 = 30, what is m 3? ee Lesson x 10 3x 7 ee Lesson 1-5. Get eady! o prepare for Lesson 5-3, do Exercises What is the slope of a line that is perpendicular to the line y = -3x + 4? 46. Line / is a horizontal line. What is the slope of a line perpendicular to /? 47. escribe the line x = 5. ee Lesson 3-8. Lesson 5-2 erpendicular and ngle isectors 299

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

8.2 Angle Bisectors of Triangles

8.2 Angle Bisectors of Triangles 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

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

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

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

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

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

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles... Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

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

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

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

4.7 Triangle Inequalities

4.7 Triangle Inequalities age 1 of 7 4.7 riangle Inequalities Goal Use triangle measurements to decide which side is longest and which angle is largest. he diagrams below show a relationship between the longest and shortest sides

More information

Name Period 11/2 11/13

Name Period 11/2 11/13 Name Period 11/2 11/13 Vocabulary erms: ongruent orresponding Parts ongruency statement Included angle Included side GOMY UNI 6 ONGUN INGL HL Non-included side Hypotenuse Leg 11/5 and 11/12 eview 11/6,,

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

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

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.

More information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts: Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

More 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

Properties of Rhombuses, Rectangles, and Squares

Properties of Rhombuses, Rectangles, and Squares 8.4 roperties of hombuses, ectangles, and quares efore You used properties of parallelograms. Now You will use properties of rhombuses, rectangles, and squares. Why? o you can solve a carpentry problem,

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

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

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

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

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

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

Chapter 1: Essentials of Geometry

Chapter 1: Essentials of Geometry Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

More 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

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

GEOMETRY OF THE CIRCLE

GEOMETRY OF THE CIRCLE HTR GMTRY F TH IRL arly geometers in many parts of the world knew that, for all circles, the ratio of the circumference of a circle to its diameter was a constant. Today, we write d 5p, but early geometers

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

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

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

Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Name: lass: _ ate: _ I: SSS Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Given the lengths marked on the figure and that bisects E, use SSS to explain

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

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 Module 4 Unit 2 Practice Exam

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

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

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

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

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

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

Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A

More 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

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

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram

More information

Perpendicular and Angle Bisectors Quiz

Perpendicular and Angle Bisectors Quiz Name: lass: ate: I: Perpendicular and ngle isectors Quiz Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the measures and. a. = 6.4, = 4.6 b. = 4.6,

More information

GEOMETRY FINAL EXAM REVIEW

GEOMETRY FINAL EXAM REVIEW GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.

More 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

Mathematics Geometry Unit 1 (SAMPLE)

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

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

Unknown Angle Problems with Inscribed Angles in Circles

Unknown Angle Problems with Inscribed Angles in Circles : Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central

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

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

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

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

Geo - CH6 Practice Test

Geo - CH6 Practice Test Geo - H6 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the measure of each exterior angle of a regular decagon. a. 45 c. 18 b. 22.5

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

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

Geometry Course Summary Department: Math. Semester 1

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

More information

Relationships Within Triangles

Relationships Within Triangles 6 Relationships Within Triangles 6.1 erpendicular and ngle isectors 6. isectors of Triangles 6.3 Medians and ltitudes of Triangles 6.4 The Triangle Midsegment Theorem 6.5 Indirect roof and Inequalities

More information

pair of parallel sides. The parallel sides are the bases. The nonparallel sides are the legs.

pair of parallel sides. The parallel sides are the bases. The nonparallel sides are the legs. age 1 of 5 6.5 rapezoids Goal Use properties of trapezoids. trapezoid is a quadrilateral with eactly one pair of parallel sides. he parallel sides are the bases. he nonparallel sides are the legs. leg

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

POTENTIAL REASONS: Definition of Congruence:

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

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

(n = # of sides) One interior angle:

(n = # of sides) One interior angle: 6.1 What is a Polygon? Regular Polygon- Polygon Formulas: (n = # of sides) One interior angle: 180(n 2) n Sum of the interior angles of a polygon = 180 (n - 2) Sum of the exterior angles of a polygon =

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

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

Geometry Chapter 1 Review

Geometry Chapter 1 Review Name: lass: ate: I: Geometry hapter 1 Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name two lines in the figure. a. and T c. W and R b. WR and

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

1.7 Find Perimeter, Circumference,

1.7 Find Perimeter, Circumference, .7 Find Perimeter, Circumference, and rea Goal p Find dimensions of polygons. Your Notes FORMULS FOR PERIMETER P, RE, ND CIRCUMFERENCE C Square Rectangle side length s length l and width w P 5 P 5 s 5

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

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

More information

Definitions, Postulates and Theorems

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

More information

Geometry 1. Unit 3: Perpendicular and Parallel Lines

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

More information

New Vocabulary inscribed angle. At the right, the vertex of &C is on O, and the sides of &C are chords of the circle. &C is an

New Vocabulary inscribed angle. At the right, the vertex of &C is on O, and the sides of &C are chords of the circle. &C is an -. lan - Inscribed ngles bjectives o find the measure of an inscribed angle o find the measure of an angle formed by a tangent and a chord amples Using the Inscribed ngle heorem Using orollaries to Find

More information

Exploring Spherical Geometry

Exploring Spherical Geometry Exploring Spherical Geometry Introduction The study of plane Euclidean geometry usually begins with segments and lines. In this investigation, you will explore analogous objects on the surface of a sphere,

More information

Grade 4 - Module 4: Angle Measure and Plane Figures

Grade 4 - Module 4: Angle Measure and Plane Figures Grade 4 - Module 4: Angle Measure and Plane Figures Acute angle (angle with a measure of less than 90 degrees) Angle (union of two different rays sharing a common vertex) Complementary angles (two angles

More 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 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 Distance from a Point to a Line

The Distance from a Point to a Line : Student Outcomes Students are able to derive a distance formula and apply it. Lesson Notes In this lesson, students review the distance formula, the criteria for perpendicularity, and the creation of

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

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

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

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than

More information

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

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

More information

This is a tentative schedule, date may change. Please be sure to write down homework assignments daily.

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

More information

ABC is the triangle with vertices at points A, B and C

ABC is the triangle with vertices at points A, B and C Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

More information

1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z

1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z eometry o T ffix tudent abel ere tudent ame chool ame istrict ame/ ender emale ale onth ay ear ate of irth an eb ar pr ay un ul ug ep ct ov ec ast ame irst ame erformance ased ssessment lace the tudent

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

Geometry: 2.1-2.3 Notes

Geometry: 2.1-2.3 Notes Geometry: 2.1-2.3 Notes NAME 2.1 Be able to write all types of conditional statements. Date: Define Vocabulary: conditional statement if-then form hypothesis conclusion negation converse inverse contrapositive

More information

Tangents and Chords Off On a Tangent

Tangents and Chords Off On a Tangent SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite circle is the set of all points in a plane

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

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points? GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

More information

Geometry, Final Review Packet

Geometry, Final Review Packet Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a

More information

Goal Find angle measures in triangles. Key Words corollary. Student Help. Triangle Sum Theorem THEOREM 4.1. Words The sum of the measures of EXAMPLE

Goal Find angle measures in triangles. Key Words corollary. Student Help. Triangle Sum Theorem THEOREM 4.1. Words The sum of the measures of EXAMPLE Page of 6 4. ngle Measures of Triangles Goal Find angle measures in triangles. The diagram below shows that when you tear off the corners of any triangle, you can place the angles together to form a straight

More information

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

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.

More information

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period 1 2 3 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,

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

Selected practice exam solutions (part 5, item 2) (MAT 360)

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

More information

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

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

More information

4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem. 14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

More information

Geometry 8-1 Angles of Polygons

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

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