NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 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. What sets this lesson apart is that necessary 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 (7 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© β₯ π«π¬ and π©πͺ β₯ π¬π. Prove that π = π. (Hint: Extend π©πͺ and π¬π«.) PROOF: π¨π© β₯ π«π¬, π©πͺ β₯ π¬π Given 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 In the previous lesson, you used deductive reasoning with labeled diagrams to prove specific conjectures. What is different about the proof above? Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 83 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 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© β₯ πͺπ«. Prove: π = π + π. Discussion (8 minutes) Students explore different ways to add auxiliary lines (construction) to the same diagram. Discussion Here is one possibility: Given: π¨π© β₯ πͺπ«. Prove: π = π + π. Extend the transversal as shown by the dotted line in the diagram. Label angle measures π and π, as shown. What do you know about π and π? About π and π? How does this help you? Write a proof using the auxiliary segment drawn in the diagram to the right. MP.7 π¨π© β₯ πͺπ« Given. π=π+π The 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 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 Another possibility appears here: Given: π¨π© β₯ πͺπ«. Prove: π = π + π. Draw a segment parallel to π¨π© through the vertex of the angle measuring π degrees. This divides the angle into two parts as shown. Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 84 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY What do you know about π and π? They are equal since they are corresponding angles of parallel lines crossed by a transversal. About π and 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© β₯ πͺπ« Given π=π If parallel lines are cut by a transversal, the corresponding angles are equal. π=π If parallel lines are cut by a transversal, the corresponding angles are equal. π=π+π Angle addition postulate π =π+π Substitution property of equality Examples (24 minutes) Examples 1. 2. In the figure to the right, π¨π© β₯ πͺπ« and π©πͺ β₯ π«π¬. Prove that πβ π¨π©πͺ = πβ πͺπ«π¬. (Is an auxiliary segment necessary?) π¨π© β₯ πͺπ«, π©πͺ β₯ π«π¬ Given πβ π¨π©πͺ = πβ π©πͺπ« If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. πβ π©πͺπ« = πβ πͺπ«π¬ If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. πβ π¨π©πͺ = πβ πͺπ«π¬ Transitive property In the figure to the right, π¨π© β₯ πͺπ« and π©πͺ β₯ π«π¬. Prove that π + π = πππ. c° Label π°. ππ β₯ ππ, ππ β₯ ππ Given 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 same-side interior angles are supplementary. π + π = πππ Substitution property of equality Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 85 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY In the figure to the right, prove that π = π + π + π. 3. Label π and π. π =π+π The exterior angle of a triangle equals the sum of the two interior opposite angles. π = π+π The exterior angle of a triangle equals the sum of the two interior opposite angles. π = π+π+π Substitution property of equality π π° Closing (1 minute) ο§ How do auxiliary lines help solve for unknown angles? οΊ Sometimes extending a segment, line, or ray, or simply adding one illuminates an intermediate step in a solution to an unknown angle problem or proof. Exit Ticket (5 minutes) Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 86 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 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 to the right, Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· . Prove that π° = π°. C 2. bo A ao B D Prove πβ π = πβ π. Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 87 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 10 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions Write a proof for each question. 1. Μ Μ Μ Μ . Prove that π° = π°. Μ Μ Μ Μ β₯ πͺπ« In the figure to the right, π¨π© Write in angles π and π . 2. bo A Μ Μ Μ Μ π¨π© β₯ Μ Μ Μ Μ πͺπ« Given 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 C ao B D Prove πβ π = πβ π. Mark angles π, π, c, and π . πβ₯π Given 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 Subtraction property of equality 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 and subtraction property of equality πβ π = πβ π Substitution property of equality Problem Set Sample Solutions 1. In the figure to the right, π¨π© β₯ π«π¬ and π©πͺ β₯ π¬π. Prove that πβ π¨π©πͺ = πβ π«π¬π. Extend Μ Μ Μ Μ π«π¬ through Μ Μ Μ Μ π©πͺ, and mark the intersection with Μ Μ Μ Μ π©πͺ as π. π¨π© β₯ π«π¬, π©πͺ β₯ π¬π Given πβ π¨π©πͺ = πβ π¬ππͺ If parallel lines are cut by a transversal, then corresponding angles are equal in measure. πβ π¬ππͺ = πβ π«π¬π If parallel lines are cut by a transversal, then corresponding angles are equal in measure. πβ π¨π©πͺ = πβ π«π¬π Transitive property Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 88 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 10 M1 GEOMETRY 2. In the figure to the right, π¨π© β₯ πͺπ«. Prove that πβ π¨π¬πͺ = π° + π°. Μ Μ Μ Μ . Μ Μ Μ Μ and πͺπ« Draw a line through π¬ parallel to π¨π© Add point π. π¨π© β₯ πͺπ« Given πβ π©π¨π¬ = πβ π¨π¬π If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. πβ π«πͺπ¬ = πβ ππ¬πͺ If parallel lines are cut by a transversal, then alternate interior angles are congruent equal in measure. πβ π¨π¬πͺ = π° + π° Angle addition postulate Lesson 10: Unknown Angle ProofsβProofs with Constructions This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 89 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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