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Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 10: Unknown Angle ProofsβProofs with Constructions Student Outcome ο§ Students write unknown angle proofs involving auxiliary lines. Lesson Notes On the second day of unknown angle proofs, students incorporate the use of constructions, specifically auxiliary lines, to help them solve problems. In this lesson, students refer to the same list of facts they have been working with in the last few lessons. The aspect that sets this lesson apart is thatnecessary information in the diagram may not be apparent without some modification. One of the most common uses for an auxiliary line is in diagrams where multiple sets of parallel lines exist. Encourage students to mark up diagrams until the necessary relationships for the proof become more obvious. Classwork Opening Exercise (6 minutes) Review the Problem Set from Lesson 9. Then, the whole class works through an example of a proof requiring auxiliary lines. Opening Exercise In the figure to the right, π¨π© || π«π¬ and π©πͺ || π¬π. Prove that π = π. (Hint: Extend π©πͺ and π¬π«.) Proof: π=π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π=π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π=π Transitive Property Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 78 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY In the previous lesson, you used deductive reasoning with labeled diagrams to prove specific conjectures. What is different about the proof above? Drawing or extending segments, lines, or rays (referred to as auxiliary lines) is frequently useful in demonstrating steps in the deductive reasoning process. Once π©πͺ and π¬π« were extended, it was relatively simple to prove the two angles congruent based on our knowledge of alternate interior angles. Sometimes there are several possible extensions or additional lines that would work equally well. For example, in this diagram, there are at least two possibilities for auxiliary lines. Can you spot them both? Given: π¨π© || πͺπ«. Prove: π = π + π. Discussion (7 minutes) Students explore different ways to add auxiliary lines (construction) to the same diagram. Discussion Here is one possibility: Given: π¨π© || πͺπ«. Prove: π = π + π. Extend the transversal as shown by the dotted line in the diagram. Label angle measuresπand π, as shown. What do you know about π and π? About πand π? How does this help you? Write a proof using the auxiliary segment drawn in the diagram to the right. MP. 7 π=π+π Exterior angle of a triangle equals the sum of the two interior opposite angles π=π If parallel lines are cut by a transversal, then corresponding angles are equal in measure π=π Vertical angles are equal in measure π =π+π Substitution Property of Equality Another possibility appears here: Given: π¨π© || πͺπ«. Prove:π = π + π. Draw a segment parallel to π¨π© through the vertex of the anglemeasuring Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 79 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY π degrees. This divides it into angles two partsas shown. What do you know about πand π? They are equal since they are corresponding angles of parallel lines crossed by a transversal. About πand π? How does this help you? They are also equal in measure since they are corresponding angles of parallel lines crossed by a transversal. Write a proof using the auxiliary segment drawn in this diagram. Notice how this proof differs from the one above. π=π If parallel lines are cut by a transversal, then corresponding angles are equal in measure π=π If parallel lines are cut by a transversal, then corresponding angles are equal in measure π=π+π Angle addition postulate π =π+π Substitution Property of Equality Examples (25 minutes) Examples 1. In the figure at the right, π¨π© || πͺπ« and π©πͺ || π«π¬. Prove that ποπ¨π©πͺ = ποπͺπ«π¬. (Where will you draw an auxiliary segment?) ποπ¨π©πͺ = ποπ©πͺπ« If parallel lines are cut by a transversal, then alternate interior angles are equal in measure ποπ©πͺπ« = ποπͺπ«π¬ If parallel lines are cut by a transversal, then alternate interior angles are equal in measure ποπ¨π©πͺ = ποπͺπ«π¬ Transitive Property c° 2. In the figure at the right, π¨π© || πͺπ« and π©πͺ || π«π¬. Prove that π + π = πππ. Label π. π=π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π + π = πππ If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π + π = πππ Substitution Property of Equality Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 80 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 3. In the figure at the right, prove that π = π + π + π. Z z Label πand π. π =π+π Exterior angle of a triangle equals the sum of the two interior opposite angles π = π+π Exterior angle of a triangle equals the sum of the two interior opposite angles π = π+π+π Substitution Property of Equality Exit Ticket (5 minutes) Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 81 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name ___________________________________________________ Date____________________ Lesson 10: Unknown Angle ProofsβProofs with Constructions Exit Ticket Write a proof for each question. 1. In the figure at the right, Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· . Prove that π = π. bo A C 2. ao B D Prove πβ π = πβ π. Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 82 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions Write a proof for each question. 1. In the figure at the right, Μ Μ Μ Μ π¨π© β₯ Μ Μ Μ Μ πͺπ«. Prove that π = π. bo A C ao B D Write in angles πand π . 2. π=π Vertical angles are equal in measure π=π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π =π Vertical angles are equal in measure π=π Substitution Property of Equality Prove πβ π = πβ π. Mark angles a, b, c, and d. πβ π + πβ π = πβ π + πβ π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure πβ π = πβ π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure πβ π = πβ π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure πβ π + πβ π = πβ π + πβ π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure πβ π = πβ π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure πβ π = πβ π Substitution Property of Equality πβ π = πβ π Substitution Property of Equality Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 83 Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Problem Set Sample Solutions 1. In the figure at the right, π¨π© || π«π¬ and π©πͺ || π¬π. Prove that ποπ¨π©πͺ = ποπ«π¬π. Extend DE through BC, and mark the intersection with BC as Z. 2. ποπ¨π©πͺ = ποπ¬ππͺ If parallel lines are cut by a transversal, then corresponding angles are equal in measure ποπ¬ππͺ = ποπ«π¬π If parallel lines are cut by a transversal, then corresponding angles are equal in measure ποπ¨π©πͺ = ποπ«π¬π Transitive Property In the figure at the right, π¨π© || πͺπ«. Prove that ποπ¨π¬πͺ = π° + π°. Μ Μ Μ Μ ; Μ Μ Μ Μ and πͺπ« Draw in line through E parallel to π¨π© add point F. ποπ©π¨π¬ = ποπ¨π¬π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure ποπ«πͺπ¬ = ποππ¬πͺ If parallel lines are cut by a transversal, then alternate interior angles are equal in measure ποπ¨π¬πͺ = π° + π° Angle addition postulate Lesson 10: Date: ©2013CommonCore,Inc. Some rights reserved.commoncore.org Unknown Angle ProofsβProofs with Constructions 4/29/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 84