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Lesson 5: Inscribed Angle Theorem and Its Applications NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY Lesson 5: 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 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 Lesson 5 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 Lesson 5. This lesson only includes inscribed angles and central angles that are acute or right. Obtuse angles are not studied until Lesson 7. Opening Exercise and Examples 1–2 are the complete proof of the inscribed angle theorem (central angle version). Classwork Opening Exercise (7 minutes) Lead students through a discussion of the Opening Exercise (an adaptation of Lesson 4 Scaffolding: 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 Include a diagram in which students to understand why the Opening Exercise supports but is not a complete proof of point 퐵 and angle 퐴퐵퐶 are the inscribed angle theorem and then to make diagrams of the remaining cases, which are already drawn. addressed in Examples 1–2 and Exercise 1. Opening Exercise a. 푨 and 푪 are points on a circle with center 푶. i. Draw a point 푩 on the circle so that 푨푩̅̅̅̅ is a diameter. Then draw the angle 푨푩푪. ii. What angle in your diagram is an inscribed angle? ∠푨푩푪 iii. What angle in your diagram is a central angle? ∠푨푶푪 Lesson 5: Inscribed Angle Theorem and Its Applications 53 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY iv. What is the intercepted arc of ∠푨푩푪? 푨푪̂ v. What is the intercepted arc of ∠푨푶푪 ? 푨푪̂ b. The measure of the inscribed angle 푨푪푫 is 풙, and the measure of the central angle 푪푨푩 is 풚. Find 풎∠푪푨푩 in terms of 풙. We are given that 풎∠푨푪푫 is 풙. We know that 푨푩 = 푨푪 = 푨푫, so △ 푪푨푫 is an isosceles triangle, which means that 풎∠푨푫푪 is also 풙. The sum of the angles of a triangle is ퟏퟖퟎ˚, so 풎∠푪푨푫 = ퟏퟖퟎ˚ − ퟐ풙. ∠푪푨푫 and ∠푪푨푩 are supplementary meaning that 풎∠푪푨푩 = ퟏퟖퟎ˚ − (ퟏퟖퟎ˚ − ퟐ풙) = ퟐ풙; therefore, 풚 = ퟐ풙. Relevant Vocabulary INSCRIBED ANGLE THEOREM (as it is 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 Scaffolding: the inscribed angle theorem. Then highlight the fact that one case was proved but not all Post drawings of each case as cases of the inscribed angle theorem (the case in which a side of the angle passes through they are studied in class as well the center of the circle) were proved. Sketch drawings of various cases to set up; for as the definitions of inscribed instance, Example 1 could be the inside case, and Example 2 could be the outside case. angles, central angles, and Consider sketching them in a place where they can be referred to throughout the class. intercepted arcs. Lesson 5: Inscribed Angle Theorem and Its Applications 54 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY . What do you notice that is the same or different about each of these pictures? MP.7 Answers will vary. What arc is intercepted by ∠퐴퐵퐶? 퐴퐶̂ . How do you know this arc is intercepted by ∠퐴퐵퐶? The endpoints of the arc (퐴 and 퐶) 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. Does the Opening Exercise satisfy the conditions of the inscribed angle theorem? Yes. ∠퐴퐵퐶 is an inscribed angle and ∠퐴푂퐶 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 Exercise? That the measure of ∠퐴푂퐶 is twice the measure of ∠퐴퐵퐶. This is because △ 푂퐵퐶 is an isosceles triangle whose legs are radii. The base angles satisfy 푚∠퐵 = 푚∠퐶 = 푥. So, 푦 = 2푥 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 Exercise 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 ∠퐴퐵퐶 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 Exercise only shows 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. Lesson 5: Inscribed Angle Theorem and Its Applications 55 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M5 GEOMETRY . There is one more case, when 퐵 is on the minor arc between 퐴 and 퐶 instead of the major arc. If students ask where the 푥 and 푦 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 Scaffolding: the circle is inside or outside the inscribed angle and the inscribed angle is acute. Both use Before doing Examples 1 and 2, similar computations based on the Opening Exercise. Example 1 is easier to see than or instead of going through the Example 2. For Example 2, consider letting students figure out the diagram on their own, examples, teachers could do an but then go through the proof as a class. exploratory activity using the Go over proofs of Examples 1–2 with the case when ∠퐴퐵퐶 is acute. If a student draws 퐵 triangles shown in Exercise 3 so that ∠퐴퐵퐶 is obtuse, save the diagram for later when doing Lesson 7. parts (a), (b), and (c). Have students measure the central Note that the diagrams for the Opening Exercise as well as Examples 1–2 have been and inscribed angles, labeled so that in each diagram, 푂 is the center, ∠퐴퐵퐶 is an inscribed angle, and ∠퐴푂퐶 is intercepting the same arc with a central angle. This consistency highlights parallels between the computations in the a protractor, then log the three cases. measurements in a table, and look for a pattern. Example 1 푨 and 푪 are points on a circle with center 푶. a. What is the intercepted arc of ∠푪푶푨? Color it red. 푨푪̂ b. Draw triangle 푨푶푪. What type of triangle is it? Why? An isosceles triangle because 푶푪 = 푶푨 (they are radii of the same circle). Lesson 5: Inscribed Angle Theorem and Its Applications 56 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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