# Inscribed Angle Theorem and Its Applications

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 : 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 cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure. Lesson Notes introduces but does not finish the inscribed angle theorem. The statement of the inscribed angle theorem in this lesson should be only in terms of the measures of central angles and inscribed angles, not the angle measures of intercepted arcs. The measure of the inscribed angle is deduced and the central angle is given, not the other way around in. This lesson only includes inscribed angles and central angles that are acute or right. Obtuse angles will not be studied until Lesson 7. Opening Exercises 1 2 and Examples 1 2 are the complete proof of the inscribed angle theorem (central angle version). Classwork Opening Exercises (7 minutes) Lead students through a discussion of the Opening Exercises (an adaptation of Lesson 4 Problem Set 6), and review terminology, especially intercepted arc. Knowing the definition of intercepted arc is critical for understanding this and future lessons. The goal is for students to understand why the Opening Exercises support but are not a complete proof of the inscribed angle theorem, and then to make diagrams of the remaining cases, which are addressed in Examples 1 2 and Exercise 1. Scaffolding: Include a diagram in which point and angle are already drawn. Opening Exercises 1. and are points on a circle with center. a. Draw a point on the circle so that is a diameter. Then draw the angle. b. What angle in your diagram is an inscribed angle? c. What angle in your diagram is a central angle? : 52

2 d. What is the intercepted arc of? Minor arc e. What is the intercepted arc of? Minor arc 2. The measure of the inscribed angle is, and the measure of the central angle is. Find in terms of. We are given =, and we know that = =, so is an isosceles triangle, meaning is also. The sum of the angles of a triangle is, so =. and are supplementary meaning that = ( ) = ; therefore, =. Relevant Vocabulary INSCRIBED ANGLE THEOREM (as it will be stated in Lesson 7): The measure of an inscribed angle is half the angle measure of its intercepted arc. INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point. ARC INTERCEPTED BY AN ANGLE: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc; in particular, the arc intercepted by a right angle is the semicircle in the interior of the angle. Exploratory Challenge (10 minutes) Review the definition of intercepted arc, inscribed angle, and central angle and then state the inscribed angle theorem. Then highlight the fact that we proved one case but not all cases of the inscribed angle theorem (the case in which a side of the angle passes through the center of the circle). Sketch drawings of various cases to set up; for instance, Example 1 could be the inside case and Example 2 could be the outside case. You may want to sketch them in a place where you can refer to them throughout the class. Scaffolding: Post drawings of each case as they are studied in the class as well as the definitions of inscribed angles, central angles, and intercepted arcs. : 53

3 MP.7 What do you notice that is the same or different about each of these pictures? Answers will vary. What arc is intercepted by? The minor arc How do you know this arc is intercepted by? The endpoints of the arc ( and ) lie on the angle. All other points of the arc are inside the angle. Each side of the angle contains one endpoint of the arc. THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Today we are going to talk about the inscribed angle theorem. It says the following: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Do the Opening Exercises satisfy the conditions of the inscribed angle theorem? Yes. is an inscribed angle and is a central angle both of which intercept the same arc. What did we show about the inscribed angle and central angle with the same intersected arc in the Opening Exercises? That the measure of is twice the measure of. This is because is an isosceles triangle whose legs are radii. The base angles satisfy = =. So, = 2 because the exterior angle of a triangle is equal to the sum of the measures of the opposite interior angles. Do the conclusions of the Opening Exercises match the conclusion of the inscribed angle theorem? Yes. Does this mean we have proven the inscribed angle theorem? No. The conditions of the inscribed angle theorem say that could be any inscribed angle. The vertex of the angle could be elsewhere on the circle. How else could the diagram look? Students may say that the diagram could have the center of the circle be inside, outside, or on the inscribed angle. The Opening Exercises only show the case when the center is on the inscribed angle. (Discuss with students that the precise way to say inside the angle is to say in the interior of the angle. ) We still need to show the cases when the center is inside and outside the inscribed angle. : 54

4 There is one more case, when is on the minor arc between and instead of the major arc. If students ask where the and are in this diagram, say we will find out in Lesson 7. Examples 1 2 (10 minutes) These examples prove the second and third case scenarios the case when the center of the circle is inside or outside the inscribed angle and the inscribed angle is acute. Both use similar computations based on the Opening Exercises. Example 1 is easier to see than Example 2. For Example 2, you may want to let students figure out the diagram on their own, but then go through the proof as a class. Go over proofs of Examples 1 2 with the case when angle is acute. If a student draws so that angle is obtuse, save the diagram for later when doing Lesson 7. Note that the diagrams for the Opening Exercises as well as Examples 1 2 have been labeled so that in each diagram, is the center, is an inscribed angle, and is a central angle. This consistency highlights parallels between the computations in the three cases. Example 1 and are points on a circle with center. Scaffolding: Before doing Examples 1 and 2, or instead of going through the examples, teachers could do an exploratory activity using the triangles shown in Exercise 3 parts (a), (b), and (c). Have students measure the central and inscribed angles, intercepting the same arc with a protractor, then log the measurements in a table, and look for a pattern. a. What is the intercepted arc of? Color it red. Minor arc b. Draw triangle. What type of triangle is it? Why? An isosceles triangle because = (they are radii of the same circle). : 55

5 c. What can you conclude about and? Why? They are equal because base angles of an isosceles triangle are equal in measure. d. Draw a point on the circle so that is in the interior of the inscribed angle. The diagram should resemble the inside case of the discussion diagrams. e. What is the intercepted arc of? Color it green. Minor arc f. What do you notice about? It is the same arc that was intercepted by the central angle. g. Let the measure of be and the measure of be. Can you prove that =? (Hint: Draw the diameter that contains point.) Let be a diameter. Let,,, and be the measures of,,, and, respectively. We can express and in terms of these measures: = +, and = +. By the Opening Exercises, = and =. Thus, =. h. Does your conclusion support the inscribed angle theorem? Yes, even when the center of the circle is in the interior of the inscribed angle, the measure of the inscribed angle is equal to half the measure of the central angle that intercepts the same arc. i. If we combine the Opening Exercises and this proof, have we finished proving the inscribed angle theorem? No. We still have to prove the case where the center is outside the inscribed angle. Example 2 and are points on a circle with center. a. Draw a point on the circle so that is in the exterior of the inscribed angle. The diagram should resemble the outside case of the discussion diagrams. : 56

6 b. What is the intercepted arc of? Color it yellow. Minor arc c. Let the measure of be and the measure of be. Can you prove that =? (Hint: Draw the diameter that contains point.) Let be a diameter. Let,,, and be the measures of,,, and, respectively. We can express and in terms of these measures: = and =. By the Opening Exercises, = and =. Thus, =. d. Does your conclusion support the inscribed angle theorem? Yes, even when the center of the circle is in the exterior of the inscribed angle, the measure of the inscribed angle is equal to half the measure of the central angle that intercepts the same arc. e. Have we finished proving the inscribed angle theorem? We have shown all cases of the inscribed angle theorem (central angle version). We do have one more case to study in Lesson 7, but it is ok not to mention it here. The last case is when the location of is on the minor arc between and. Ask students to summarize the results of these theorems to each other before moving on. Exercises 1 5 (10 minutes) Exercises are listed in order of complexity. Students do not have to do all problems. Problems can be specifically assigned to students based on ability. Exercises Find the measure of the angle with measure. a. = b. = c. = : 57

7 d. = e. f. = = = 2. Toby says is a right triangle because =. Is he correct? Justify your answer. Toby is not correct. The =. is inscribed in the same arc as the central angle, so it has a measure of. This means that = because the sum of the angles of a triangle is. = since they are vertical angles, so the triangle is not right. 3. Let s look at relationships between inscribed angles. a. Examine the inscribed polygon below. Express in terms of and in terms of. Are the opposite angles in any quadrilateral inscribed in a circle supplementary? Explain. = ; =. The angles are supplementary. b. Examine the diagram below. How many angles have the same measure, and what are their measures in terms of? Let and be the points on the circle that the original angles contain. All the angles intercepting the minor arc between and have measure, and the angles intercepting the major arc between and measure. : 58

8 4. Find the measures of the labeled angles. a. b. =, = = c. d. = =, = e. f. = = : 59

9 Closing (3 minutes) With a partner, do a 30-second Quick Write of everything that we learned today about the inscribed angle theorem. Today we began by revisiting the Problem Set from yesterday as the key to the proof of a new theorem, the inscribed angle theorem. The practice we had with different cases of the proof allowed us to recognize the many ways that the inscribed angle theorem can show up in unknown angle problems. We then solved some unknown angle problems using the inscribed angle theorem combined with other facts we knew before. Have students add the theorems in the Lesson Summary to their graphic organizer on circles started in Lesson 2 with corresponding diagrams. Lesson Summary THEOREMS: 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. CONSEQUENCE OF INSCRIBED ANGLE THEOREM: Inscribed angles that intercept the same arc are equal in measure. Relevant Vocabulary INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point. INTERCEPTED ARC: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc. An angle inscribed in a circle intercepts exactly one arc; in particular, the arc intercepted by a right angle is the semicircle in the interior of the angle. Exit Ticket (5 minutes) : 60

10 Name Date : Exit Ticket The center of the circle below is. If angle has a measure of 15 degrees, find the values of and. Explain how you know. : 61

11 Exit Ticket Sample Solutions The center of the circle below is. If angle has a measure of degrees, find the values of and. Explain how you know. =. Triangle is isosceles, so base angles and are congruent. = =. =. is a central angle inscribed in the same arc as inscribed. So, =. Problem Set Sample Solutions Problems 1 2 are intended to strengthen students understanding of the proof of the inscribed angle theorem. The other problems are applications of the inscribed angle theorem. Problems 3 5 are the most straightforward of these, followed by Problem 6, then Problems 7 9, which combine use of the inscribed angle theorem with facts about triangles, angles, and polygons. Finally, Problem 10 combines all the above with the use of auxiliary lines in its proof. For Problems 1 8, find the value of = = : 62

12 3. 4. = = = = = = : 63

13 9. a. The two circles shown intersect at and. The center of the larger circle,, lies on the circumference of the smaller circle. If a chord of the larger circle,, cuts the smaller circle at, find and. = ; = b. How does this problem confirm the inscribed angle theorem? is a central angle of the larger circle and is double, the inscribed angle of the larger circle. is inscribed in the smaller circle and equal in measure to, which is also inscribed in the smaller circle. 10. In the figure below, and intersect at point. Prove: + = () PROOF: Join. = ( ) = ( ) In, + = ( ) + ( ) = + = () ; ; ;,, : 64

### 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

### The Inscribed Angle Alternate A Tangent Angle

Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is

### 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

### Arc Length and Areas of Sectors

Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.

### 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

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

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

### The 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

### MATH STUDENT BOOK. 8th Grade Unit 6

MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular

### 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

### Lesson 6: Drawing Geometric Shapes

Student Outcomes Students use a compass, protractor, and ruler to draw geometric shapes based on given conditions. Lesson Notes The following sequence of lessons is based on standard 7.G.A.2: Draw (freehand,

### Lines That Pass Through Regions

: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe

### 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.

### 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

### Circle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.

Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,

### 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

### 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:

### 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

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### 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

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### 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

### **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle.

Geometry Week 7 Sec 4.2 to 4.5 section 4.2 **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle. Protractor Postulate:

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### 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,

### 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

### 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

### A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.

hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.

### BC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because

150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,

### 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

### 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,

### 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,

### 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

### Circle theorems CHAPTER 19

352 CHTE 19 Circle theorems In this chapter you will learn how to: use correct vocabulary associated with circles use tangent properties to solve problems prove and use various theorems about angle properties

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

### Featured Mathematical Practice: MP.5. Use appropriate tools strategically. MP.6. Attend to precision.

Domain: Geometry 4.G Mathematical Content Standard: 1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures.

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,

### circumscribed circle Vocabulary Flash Cards Chapter 10 (p. 539) Chapter 10 (p. 530) Chapter 10 (p. 538) Chapter 10 (p. 530)

Vocabulary Flash ards adjacent arcs center of a circle hapter 10 (p. 539) hapter 10 (p. 530) central angle of a circle chord of a circle hapter 10 (p. 538) hapter 10 (p. 530) circle circumscribed angle

### The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons.

Interior Angles of Polygons The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. The sum of the measures of the interior angles of a triangle

### 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

### 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

### 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,

### 1 of 69 Boardworks Ltd 2004

1 of 69 2 of 69 Intersecting lines 3 of 69 Vertically opposite angles When two lines intersect, two pairs of vertically opposite angles are formed. a d b c a = c and b = d Vertically opposite angles are

### 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

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### 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.

### (a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

### 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/).

### Maximizing Angle Counts for n Points in a Plane

Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF

### 10.5 and 10.6 Lesson Plan

Title: Secants, Tangents, and Angle Measures 10.5 and 10.6 Lesson Plan Course: Objectives: Reporting Categories: Related SOL: Vocabulary: Materials: Time Required: Geometry (Mainly 9 th and 10 th Grade)

### CIRCLE DEFINITIONS AND THEOREMS

DEFINITIONS Circle- The set of points in a plane equidistant from a given point(the center of the circle). Radius- A segment from the center of the circle to a point on the circle(the distance from the

### Lesson 6.3: Arcs and Angles

Lesson 6.3: Arcs and Angles In this lesson you will: make conjectures about inscribed angles in a circle investigate relationships among the angles in a cyclic quadrilateral compare the arcs formed when

### 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:

### 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

### 1 Solution of Homework

Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,

### 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

### Geometry. Unit 6. Quadrilaterals. Unit 6

Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections

### Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

### 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

### #2. Isosceles Triangle Theorem says that If a triangle is isosceles, then its BASE ANGLES are congruent.

1 Geometry Proofs Reference Sheet Here are some of the properties that we might use in our proofs today: #1. Definition of Isosceles Triangle says that If a triangle is isosceles then TWO or more sides

### Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles

Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles IMPORTANT TERMS AND DEFINITIONS parallelogram rectangle square rhombus A quadrilateral is a polygon that has four sides. A parallelogram is

### 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

### 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.

### How Do You Measure a Triangle? Examples

How Do You Measure a Triangle? Examples 1. A triangle is a three-sided polygon. A polygon is a closed figure in a plane that is made up of segments called sides that intersect only at their endpoints,

### Unit 3: Circles and Volume

Unit 3: Circles and Volume This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,

### 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

### 6. Angles. a = AB and b = AC is called the angle BAC.

6. Angles Two rays a and b are called coterminal if they have the same endpoint. If this common endpoint is A, then there must be points B and C such that a = AB and b = AC. The union of the two coterminal

### 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

### 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.

### TImath.com. Geometry. Triangle Sides & Angles

Triangle Sides & Angles ID: 8792 Time required 40 minutes Activity Overview In this activity, students will explore side and angle relationships in a triangle. First, students will discover where the longest

### INFORMATION FOR TEACHERS

INFORMATION FOR TEACHERS The math behind DragonBox Elements - explore the elements of geometry - Includes exercises and topics for discussion General information DragonBox Elements Teaches geometry through

### 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

### Lesson 7: Drawing Parallelograms

Student Outcomes Students use a protractor, ruler, and setsquare to draw parallelograms based on given conditions. Lesson Notes In Lesson 6, students drew a series of figures (e.g., complementary angles,

### NCERT. In examples 1 and 2, write the correct answer from the given four options.

MTHEMTIS UNIT 2 GEOMETRY () Main oncepts and Results line segment corresponds to the shortest distance between two points. The line segment joining points and is denoted as or as. ray with initial point

### Set 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles

Goal: To provide opportunities for students to develop concepts and skills related to circumference, arc length, central angles, chords, and inscribed angles Common Core Standards Congruence Experiment

### Reflex Vertices A 2. A 1 Figure 2a

Reflex Vertices For any integer n 3, the polygon determined by the n points 1,..., n (1) in the plane consists of the n segments 1 2, 2 3,..., n-1 n, n 1, (2) provided that we never pass through a point

### 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,

### Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,

### Lesson 13: Angle Sum of a Triangle

Student Outcomes Students know the angle sum theorem for triangles; the sum of the interior angles of a triangle is always 180. Students present informal arguments to draw conclusions about the angle sum

### parallel lines perpendicular lines intersecting lines vertices lines that stay same distance from each other forever and never intersect

parallel lines lines that stay same distance from each other forever and never intersect perpendicular lines lines that cross at a point and form 90 angles intersecting lines vertices lines that cross

### Unit Background. Stage 1: Big Goals

Course: 7 th Grade Mathematics Unit Background Unit Title Angle Relationships, Transversals & 3-Dimensional Geometry Does the unit title reflect the standards? PCCMS Unit Reviewer (s): Unit Designer: Resources

### 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,

### CRLS Mathematics Department Geometry Curriculum Map/Pacing Guide. CRLS Mathematics Department Geometry Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page of 6 2 77.5 Unit : Tools of 5 9 Totals Always Include 2 blocks for Review & Test Activity binder, District Google How do you find length, area? 2 What are the basic tools

### Geometry Vocabulary. Created by Dani Krejci referencing:

Geometry Vocabulary Created by Dani Krejci referencing: http://mrsdell.org/geometry/vocabulary.html point An exact location in space, usually represented by a dot. A This is point A. line A straight path

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,

### Line. A straight path that continues forever in both directions.

Geometry Vocabulary Line A straight path that continues forever in both directions. Endpoint A point that STOPS a line from continuing forever, it is a point at the end of a line segment or ray. Ray A

### 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

### 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

### Vertex : is the point at which two sides of a polygon meet.

POLYGONS A polygon is a closed plane figure made up of several line segments that are joined together. The sides do not cross one another. Exactly two sides meet at every vertex. Vertex : is the point

### Inscribed (Cyclic) Quadrilaterals and Parallelograms

Lesson Summary: This lesson introduces students to the properties and relationships of inscribed quadrilaterals and parallelograms. Inscribed quadrilaterals are also called cyclic quadrilaterals. These

### Name: 22K 14A 12T /48 MPM1D Unit 7 Review True/False (4K) Indicate whether the statement is true or false. Show your work

Name: _ 22K 14A 12T /48 MPM1D Unit 7 Review True/False (4K) Indicate whether the statement is true or false. Show your work 1. An equilateral triangle always has three 60 interior angles. 2. A line segment

### 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

### Activity Set 4. Trainer Guide

Geometry and Measurement of Plane Figures Activity Set 4 Trainer Guide Int_PGe_04_TG GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 NGSSS 3.G.3.1 NGSSS 3.G.3.3 NGSSS 4.G.5.1 NGSSS 5.G.3.1 Amazing

### 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

### Ch 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and Angles [and Triangles]

h 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and ngles [and Triangles] Warm up: Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having.

### Unit 6 Geometry: Constructing Triangles and Scale Drawings

Unit 6 Geometry: Constructing Triangles and Scale Drawings Introduction In this unit, students will construct triangles from three measures of sides and/or angles, and will decide whether given conditions

### A Different Look at Trapezoid Area Prerequisite Knowledge

Prerequisite Knowledge Conditional statement an if-then statement (If A, then B) Converse the two parts of the conditional statement are reversed (If B, then A) Parallel lines are lines in the same plane