NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 M5 GEOMETRY Lesson 13: 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 tangent). Students solve a variety of missing angle problems using the inscribed angle theorem. Lesson Notes The Opening Exercise reviews and solidifies the concept of inscribed angles and their intercepted arcs. Students then extend that knowledge in the remaining examples to the limiting case of inscribed angles, one ray of the angle is tangent. The Exploratory Challenge looks at a tangent and secant intersecting on the circle. The Example uses rotations to show the connection between the angle formed by the tangent and a chord intersecting on the circle and the inscribed angle of the second arc. Students then use all of the angle theorems studied in this topic to solve missing angle problems. Classwork Scaffolding: Opening Exercise (5 minutes) . Post diagrams showing key This exercise solidifies the relationship between inscribed angles and their intercepted theorems for students to arcs. Have students complete this exercise individually and then share with a neighbor. refer to. Pull the class together to answer questions and discuss part (g). Use scaffolded questions with a targeted small Opening Exercise group such as, “What do we know about the In circle 푨, 풎푩푫̂ = ퟓퟔ°, and 푩푪̅̅̅̅ is a diameter. Find the listed measure, and explain your answer. measure of the a. 풎∠푩푫푪 intercepted arc of an ퟗퟎ°, angles inscribed in a diameter inscribed angle?” b. 풎∠푩푪푫 ퟐퟖ°, inscribed angle is half measure of intercepted arc c. 풎∠푫푩푪 ퟔퟐ°, sum of angles of a triangle is ퟏퟖퟎ° d. 풎∠푩푭푮 ퟐퟖ°, inscribed angle is half measure of intercepted arc Lesson 13: The Inscribed Angle Alternate—A Tangent Angle 170 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 13 M5 GEOMETRY e. 풎푩푪̂ ퟏퟖퟎ°, semicircle f. 풎푫푪̂ ퟏퟐퟒ°, intercepted arc is double inscribed angle g. Does ∠푩푮푫 measure ퟓퟔ°? Explain. No, the central angle of 푩푫̂ would be ퟓퟔ°. ∠푩푮푫 is not a central angle because its vertex is not the center of the circle. h. How do you think we could determine the measure of ∠푩푮푫? Answers will vary. This leads to today’s lesson. Exploratory Challenge (15 minutes) In the Lesson 12 Problem Set, students were asked to find a relationship between the measure of an arc and an angle. The point of the Exploratory Challenge is to establish the following conjecture for the class community and prove the conjecture. CONJECTURE: Let 퐴 be a point on a circle, let ⃗퐴퐵⃗⃗⃗⃗ be a tangent ray to the circle, and let 퐶 be a point on the circle such that 1 퐴퐶⃡⃗⃗⃗⃗ is a secant to the circle. If 푎 = 푚∠퐵퐴퐶 and 푏 is the angle measure of the arc intercepted by ∠퐵퐴퐶, then 푎 = 푏. 2 The Opening Exercise establishes empirical evidence toward the conjecture and helps students determine whether their reasoning on the homework may have had flaws; it can be used to see how well students understand the diagram and to review how to measure arcs. Students need a protractor and a ruler. Exploratory Challenge Diagram 1 Diagram 2 Lesson 13: The Inscribed Angle Alternate—A Tangent Angle 171 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 13 M5 GEOMETRY Examine the diagrams shown. Develop a conjecture about the relationship between 풂 and 풃. ퟏ 풂 = 풃 ퟐ Test your conjecture by using a protractor to measure 풂 and 풃. 풂 풃 Diagram 1 ퟔퟐ ퟏퟐퟒ Diagram 2 ퟑퟔ ퟕퟐ Do your measurements confirm the relationship you found in your homework? If needed, revise your conjecture about the relationship between 풂 and 풃: Now, test your conjecture further using the circle below. 풂 풃 . What did you find about the relationship between 푎 and 푏? 1 푎 = 푏. An angle inscribed between a tangent line and secant line is equal to half of the angle 2 measure of its intercepted arc. How did you test your conjecture about this relationship? Look for evidence that students recognized that the angle should be formed by a secant intersecting a tangent at the point of tangency and that they knew to measure the arc by taking its central angle. What conjecture did you come up with? Share with a neighbor. Let students discuss, and then state a version of the conjecture publicly. Lesson 13: The Inscribed Angle Alternate—A Tangent Angle 172 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 13 M5 GEOMETRY Now, we will prove your conjecture, which is stated below as a theorem. THE TANGENT-SECANT THEOREM: Let 푨 be a point on a circle, let 푨푩⃗⃗⃗⃗⃗⃗ be a tangent ray to the circle, and let 푪 be a point on the circle such that 푨푪⃡⃗⃗⃗⃗ is a secant to the circle. If 풂 = 풎∠푩푨푪 and 풃 is the angle measure of the arc intercepted by ∠푩푨푪, ퟏ 풂 = 풃 then ퟐ . Given circle 푶 with tangent 푨푩⃡⃗⃗⃗⃗ , prove what we have just discovered using what you know about the properties of a circle and tangent and secant lines. a. Draw triangle 푨푶푪. What is the measure of ∠푨푶푪? Explain. 풃°. The central angle is equal to the degree measure of the arc it intercepts. b. What is the measure of ∠푶푨푩? Explain. ퟗퟎ°. The radius is perpendicular to the tangent line at the point of tangency. c. Express the measure of the remaining two angles of triangle 푨푶푪 in terms of 풂 and explain. The angles are congruent because the triangle is isosceles. Each angle has a measure of (ퟗퟎ − 풂)° since 풎∠푶푨푪 + 풎∠푪푨푩 = ퟗퟎ°. d. What is the measure of ∠푨푶푪 in terms of 풂? Show how you got the answer. ퟏ ퟏퟖퟎ° ퟗퟎ − 풂 + ퟗퟎ − 풂 + 풃 = ퟏퟖퟎ 풃 = ퟐ풂 풂 = 풃 The sum of the angles of a triangle is , so . Therefore, or ퟐ . e. Explain to your neighbor what we have just proven. An inscribed angle formed by a secant and tangent line is half of the angle measure of the arc it intercepts. Example (5 minutes) We have shown that the inscribed angle theorem can be extended to the case when one of the angle’s rays is a tangent segment and the vertex is the point of tangency. The Example develops another theorem in the inscribed angle theorem’s family: the angle formed by the intersection of the tangent line and a chord of the circle on the circle and the inscribed angle of the same arc are congruent. This example is best modeled with dynamic Geometry software. Alternatively, the teacher may ask students to create a series of sketches that show point 퐸 moving toward point 퐴. THEOREM: Suppose ̅퐴퐵̅̅̅ is a chord of circle 퐶, and 퐴퐷̅̅̅̅ is a tangent segment to the circle at point 퐴. If 퐸 is any point other than 퐴 or 퐵 in the arc of 퐶 on the opposite side of ̅퐴퐵̅̅̅ from 퐷, then 푚∠퐵퐸퐴 = 푚∠퐵퐴퐷. Lesson 13: The Inscribed Angle Alternate—A Tangent Angle 173 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 13 M5 GEOMETRY . Draw a circle, and label it 퐶. Students draw circle 퐶. Draw a chord ̅퐴퐵̅̅̅. Students draw chord ̅퐴퐵̅̅̅. Construct a segment tangent to the circle through point 퐴, and label it 퐴퐷̅̅̅̅. Students construct tangent segment 퐴퐷̅̅̅̅. Now, draw and label point 퐸 that is between 퐴 and 퐵 but on the other side of chord ̅퐴퐵̅̅̅ from 퐷. Students draw point 퐸. Rotate point 퐸 on the circle toward point 퐴. What happens to 퐸퐵̅̅̅̅? 퐸퐵̅̅̅̅ moves closer and closer to lying on top of ̅퐴퐵̅̅̅ as 퐸 gets closer and closer to 퐴. What happens to 퐸퐴̅̅̅̅? 퐸퐴̅̅̅̅ moves closer and closer to lying on top of 퐴퐷̅̅̅̅. What happens to ∠퐵퐸퐴? ∠퐵퐸퐴 moves closer and closer to lying on top of ∠퐵퐴퐷. Does 푚∠퐵퐸퐴 change as it rotates? No, it remains the same because the intercepted arc length does not change. Explain how these facts show that 푚∠퐵퐸퐴 = 푚∠퐵퐴퐷? The measure of ∠퐵퐸퐴 does not change. The segments are just rotated, but the angle measure is conserved; regardless of where 퐸 is between 퐵 and 퐴, the same arc is intercepted as ∠퐵퐴퐷. Exercises (12 minutes) Students should work on the exercises individually and then compare answers with a neighbor. Walk around the room, and use this as a quick informal assessment.