NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M5 GEOMETRY Lesson 6: 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 angles. Lesson Notes Lesson 6 continues the work of Lesson 5 on the inscribed angle theorem. Many of the problems in Lesson 6 have chords that meet outside of the circle, and students are looking at relationships between triangles formed within circles and finding angles using their knowledge of the inscribed angle theorem and Thales’ theorem. When working on unknown angle problems, present them as puzzles to be solved. Students are to use what is known to find missing pieces of the puzzle until they find the piece asked for in the problem. Calling these puzzles instead of problems encourages students to persevere in their work and see it more as a fun activity. Classwork Scaffolding: Opening Exercise (10 minutes) . Create a Geometry Allow students to work in pairs or groups of three and work through the proof below, Axiom/Theorem wall, similar to writing their work on large paper. Some groups may need more guidance, and others a Word Wall, so students will may need this problem modeled. Call students back together as a class, and have have easy reference. Allow groups present their work. Use this as an informal assessment of student students to create colorful understanding. Compare work, and clear up misconceptions. Also, talk about designs and display their work. different strategies groups used. For example, a student draws a picture of an inscribed angle Opening Exercise and a central angle In a circle, a chord 푫푬̅̅̅̅ and a diameter 푨푩̅̅̅̅ are extended outside of the circle to meet at point intercepting the same arc and 푪. If 풎∠푫푨푬 = ퟒퟔ°, and 풎∠푫푪푨 = ퟑퟐ°, find 풎∠푫푬푨. color codes it with the angle relationship between the two noted. Students could be assigned axioms, theorems, or terms to illustrate so that all students would have work displayed. For advanced learners, present the problem from the Opening Exercise, and ask them to construct the proof without the guided steps. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 66 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 6 M5 GEOMETRY Let 풎∠푫푬푨 = 풚˚, 풎∠푬푨푩 = 풙˚ In △ 푨푩푫, 풎∠푫푩푨 = 풚˚ Reason: angles inscribed in same arc are congruent 풎∠푨푫푩 = ퟗퟎ° Reason: angle inscribed in semicircle ∴ ퟒퟔ + 풙 + 풚 + ퟗퟎ = ퟏퟖퟎ Reason: sum of angles of triangle is ퟏퟖퟎ° 풙 + 풚 = ퟒퟒ In △ 푨푪푬, 풚 = 풙 + ퟑퟐ Reason: Exterior angle of a triangle is equal to the sum of the remote interior angles 풙 + 풙 + ퟑퟐ = ퟒퟒ Reason: substitution 풙 = ퟔ 풚 = ퟑퟖ 풎∠푫푬푨 = ퟑퟖ° Exploratory Challenge (15 minutes) Display the theorem below for the class to see. Have students state the theorem in their own words. Lead students through the first part of the proof of the theorem with leading questions, and then divide the class into partner groups. Have half of the groups prove why 퐵′ cannot be outside of the circle and half of the class prove why 퐵′ cannot be inside of the circle; then as a whole class, have groups present their work and discuss. Do the following as a whole class: . THEOREM: If 퐴, 퐵, 퐵′, and 퐶 are four points with 퐵 and 퐵′ on the same side of 퐴퐶⃡ , and ∠퐴퐵퐶 and ∠퐴퐵′퐶 are congruent, then 퐴, 퐵, 퐵′, and 퐶 all lie on the same circle. State this theorem in your own words, and write it on a piece of paper. Share it with a neighbor. If we have 2 points on a circle (퐴 and 퐶), and two points between those two points on the same side (퐵 and 퐵′), and if we draw two angles that are congruent (∠퐴퐵퐶 and ∠퐴퐵′퐶), then all of the points (퐴, 퐵, 퐵′, and 퐶) lie on the same circle. Let’s start with points 퐴, 퐵, and 퐶. Draw a circle containing points 퐴, 퐵, and 퐶. Students draw a circle with points 퐴, 퐵, and 퐶 on the circle. Draw ∠퐴퐵퐶. Students draw ∠퐴퐵퐶. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 67 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 6 M5 GEOMETRY . Do we know the measure of ∠퐴퐵퐶? No. If students want to measure it, remind them that all circles drawn by their classmates are different, so we are finding a general case. Since we do not know the measure of this angle, assign it to be the variable 푥, and label your drawing. Students label the diagram. In the theorem, we are told that there is another point 퐵′. What else are we told about 퐵′? 퐵′ lies on the same side of 퐴퐶⃡ as 퐵. ∠퐴퐵퐶 ≅ ∠퐴퐵′퐶 . What are we trying to prove? Figure 1 퐵′ lies on the circle too. Let’s look at a case where 퐵′ is not on the circle. Where could 퐵′ lie? 퐵′ could lie outside of the circle or inside of the circle. Assign each half of the class an investigation: one half will examine if 퐵′ can lie outside of the circle, while the other half examines whether 퐵′ can lie inside the circle. Figure 2 The following questions can be used to guide the group examining if 퐵′ could be outside the circle: . Let’s look at the case where it lies outside of the circle. Draw 퐵′ outside of your circle, and draw ∠퐴퐵′퐶. Students draw 퐵′ and ∠퐴퐵′퐶. What is mathematically wrong with this picture? Answers will vary. We want students to see that the inscribed angle, ∠퐴퐷퐶, has a measure of 푥 since it is inscribed in the same arc as ∠퐴퐵퐶. See Figure 1. To further clarify, have students draw the △ 퐴퐵′퐶 with the inscribed segment as shown in Figure 2. Further discuss what is mathematically incorrect with the angles marked 푥 in the triangle; students should observe that the angle measures of triangles 퐴퐵′퐶 and 퐴퐷퐶 cannot both sum to 180˚. Figure 3 . What can we conclude about 퐵′? 퐵′ cannot lie outside of the circle. The following questions can be used to guide the group examining if 퐵′ could be inside the circle: . Let’s look at the case where it lies inside of the circle. Draw 퐵′ inside of your circle, and draw ∠퐴퐵′퐶. What is mathematically wrong with this picture? Figure 4 Answers will vary. Again, ∠퐴퐷퐶 circle has a measure of 푥 since it is inscribed in the same arc as ∠퐴퐵퐶. See Figure 3. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 68 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 6 M5 GEOMETRY To further clarify, have students draw the triangle △ 퐴퐵′퐶 with the inscribed segment as shown in Figure 4. Further discuss what is mathematically incorrect with the angles marked 푥 in the triangle. Once again, students should observe that triangles 퐴퐵′퐶 and 퐴퐷퐶 cannot both sum to 180˚. Circle around as groups are working, and help where necessary, leading some groups if required. Call the class back together, and allow groups to present their findings. Discuss both cases as a class. Do a 30-second Quick Write on what you have just discovered. If 퐴, 퐵, 퐵′, and 퐶 are 4 points with 퐵 and 퐵′ on the same side of 퐴퐶⃡ , and ∠퐴퐵퐶 and ∠퐴퐵′퐶 are congruent, then 퐴, 퐵, 퐵′, and 퐶 all lie on the same circle. Exercises (13 minutes) Have students work through the problems (puzzles) below in pairs or homogeneous groups of three. Some groups may need one-on-one guidance. As students complete problems, have them summarize the steps that they took to solve each problem; then post solutions at 5-minute intervals. This gives hints to the groups that are stuck and shows different methods for solving. Exercises Find the value of 풙 in each figure below, and describe how you arrived at the answer. 1. Hint: Thales’ theorem 2. 풎∠푩푬푪 = ퟗퟎ° inscribed in a semicircle 풎∠푩푨푫 = ퟏퟒퟔ˚, if parallel lines cut by a transversal, 풎∠푬푩푪 = 풎∠푬푪푩 = ퟒퟓ° base angles of an then interior angles on the same side are isosceles triangle are congruent and sum of angles supplementary. Then the 풎푩푫̂ = ퟏퟒퟔ˚, because of a triangle = ퟏퟖퟎ° ∠푩푨푫 is a central angle intercepting 푩푫̂ . Then 풎∠푬푩푪 = 풎∠푬푫푪 = ퟒퟓ° angles inscribed in the remaining arc of the circle, 푩푪푫̂ , has a measure of same arc are congruent ퟐퟏퟒ°. Then m∠푩푬푫 = ퟏퟎퟕ˚ since it is an inscribed 풙 = ퟒퟓ angle intercepting 푩푪푫̂ . The angle sum of a quadrilateral is ퟑퟔퟎ°, which means 풙 = ퟕퟑ. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 69 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds.