Lesson 7 - EngageNY
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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY Lesson 7: Solve for Unknown Angles—Transversals Student Outcomes ο§ Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation of unknown angle proofs. Lesson Notes The focus of the second day of unknown angle problems is problems with parallel lines crossed by a transversal. This lesson features one of the main theorems (facts) learned in Grade 8: 1. If two lines are cut by a transversal and corresponding angles are equal, then the lines are parallel. 2. If parallel lines are cut by a transversal, corresponding angles are equal. (This second part is often called the parallel postulate, which tells us a property that parallel lines have that cannot be deduced from the definition of parallel lines.) Of course, students probably remember these two statements as a single fact: For two lines cut by a transversal, the measures of corresponding angles are equal if and only if the lines are parallel. Decoupling these two statements from the unified statement is the work of later lessons. The lesson begins with review material from Lesson 6. In the Discussion and Examples, students review how to identify MP.7 and apply corresponding angles, alternate interior angles, and same-side interior angles. The key is to make sense of the structure within each diagram. Before moving on to the Exercises, students learn examples of how and when to use auxiliary lines. Again, the use of auxiliary lines is another opportunity for students to make connections between facts they already know and new information. The majority of the lesson involves solving problems. Gauge how often to prompt and review answers as the class progresses; check to see whether facts from Lesson 6 are fluent. Encourage students to draw in all necessary lines and congruent angle markings to help assess each diagram. The Problem Set should be assigned in the last few minutes of class. Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 61 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY Classwork Opening Exercise (4 minutes) Opening Exercise Use the diagram at the right to determine π and π. β‘ are straight lines. β‘ and πͺπ« π¨π© π = ππ π = ππ Name a pair of vertical angles: ∠π¨πΆπͺ, ∠π«πΆπ© Find the measure of ∠π©πΆπ. Justify your calculation. π∠π©πΆπ = ππ° Linear pairs form supplementary angles. Discussion (4 minutes) Review the angle facts pertaining to parallel lines crossed by a transversal. Ask students to name examples that illustrate each fact: Discussion Given line π¨π© and line πͺπ« in a plane (see the diagram below), a third line π¬π is called a transversal if it intersects β‘π¨π© at a β‘ at a single but different point. Line π¨π© and line πͺπ« are parallel if and only if the following single point and intersects πͺπ« types of angle pairs are congruent or supplementary: ο§ Corresponding angles are equal in measure. ∠π and ∠π , ∠π and ∠π, etc. ο§ Alternate interior angles are equal in measure. ∠π and ∠π, ∠π and ∠π ο§ Same-side interior angles are supplementary. ∠π and ∠π, ∠π and ∠π Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 62 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Examples (7 minutes) Students try examples based on the Discussion; review and then discuss auxiliary line. Examples 1. 2. π∠π = ππ° 3. π∠π = πππ° 4. MP.7 π∠π = ππ° 5. An auxiliary line π∠π = ππ° is sometimes useful when solving for unknown angles. In this figure, we can use the auxiliary line to find the measures of ∠π and ∠π (how?) and then add the two measures together to find the measure of ∠πΎ. What is the measure of ∠πΎ? π∠π = ππ°, π∠π = ππ°, π∠πΎ = ππ° Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 63 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exercises 1–10 (24 minutes) Students work on this set of exercises; review periodically. Exercises 1–10 In each exercise below, find the unknown (labeled) angles. Give reasons for your solutions. 1. π∠π = ππ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure. π∠π = ππ° Vertical angles are equal in measure. π∠π = πππ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary. 2. π∠π = πππ° Linear pairs form supplementary angles; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure. MP.7 3. π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. π∠π = ππ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then interior angles on the same side are supplementary. 4. π∠π = ππ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then interior angles on the same side are supplementary. 5. π∠π = πππ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary. Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 64 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 6. π∠π = πππ° Linear pairs form supplementary angles; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure. 7. π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. π∠π = ππ° Consecutive adjacent angles on a line sum to πππ°. π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. 8. MP.7 π∠π = ππ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure. 9. π∠π = ππ° Consecutive adjacent angles on a line sum to πππ°. π∠π = ππ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure. 10. π∠π = ππ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure. Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 65 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY Relevant Vocabulary Relevant Vocabulary ALTERNATE INTERIOR ANGLES: Let line π be a transversal to lines π and π such that π intersects π at point π· and intersects π at point πΈ. Let πΉ be a point on line π and πΊ be a point on line π such that the points πΉ and πΊ lie in opposite half-planes of π. Then ∠πΉπ·πΈ and ∠π·πΈπΊ are called alternate interior angles of the transversal π with respect to line π and line π. CORRESPONDING ANGLES: Let line π be a transversal to lines π and π. If ∠π and ∠π are alternate interior angles and ∠π and ∠π are vertical angles, then ∠π and ∠π are corresponding angles. Closing (1 minute) ο§ Describe the conditions that tell us when a pair of lines cut by a transversal must be parallel. οΊ Two lines intersected by a transversal must be parallel when corresponding angles are congruent, when alternate interior angles are congruent, and when same-side interior angles are supplementary. Exit Ticket (5 minutes) Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 66 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name Date Lesson 7: Solving for Unknown Angles—Transversals Exit Ticket Determine the value of each variable. π₯= π¦= π§= Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 67 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions Determine the value of each variable. π = ππ. ππ π = πππ. π π = ππ. π Problem Set Sample Solutions Find the unknown (labeled) angles. Give reasons for your solutions. 1. π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. MP.7 2. π∠π = ππ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure. π∠π = ππ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure. Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 68 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. 3. π∠π = ππ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure. MP.7 4. π∠π = ππ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary; vertical angles are equal in measure. Lesson 8: Solve for Unknown Angles—Transversals This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 69 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
