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Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 7: Solve for Unknown AnglesβTransversals 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 will be 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 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 55 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: β π¨πΆπͺ, β π«πΆπ© Find the measure of β π©πΆπ. Justify your calculation. π¦β 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 β‘ at a 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 π¨π© 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 β π and β π , β π and β π, etc. ο§ Alternate interior angles are equal in measure β π and β π, β π and β π ο§ Same-side interior angles are supplementary β π and β π, β π and β π Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 56 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 (8 minutes) Students try examples based on the Discussion; review, then discuss auxiliary line. Examples 1. 2. π¦β π = ππ° 3. π¦β π = πππ° 4. MP.7 π¦β π = ππ° 5. π¦β π = ππ° An auxiliary line is sometimes useful when solving for unknown angles. In this figure, we can use the auxiliary line to find the measures of β π and β π (how?), then add the two measures together to find the measure of β πΎ. What is the measure of β πΎ? π¦β π = ππ°, π¦β π = ππ°, π¦β πΎ = ππ° Exercises (24 minutes) Students work on this set of exercises; review periodically. Exercises In each exercise below, find the unknown (labeled) angles. Give reasons for your solutions. π¦β π = ππ°, If parallel lines are cut by a transversal, then corresponding angles are equal in measure 1. MP.7 π¦β π = ππ°, Vertical angles are equal in measure π¦β π = πππ°, If parallel lines are cut by a transversal, then interior angles on the same side are supplementary Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 57 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 2. π¦β π = πππ°, Linear pairs form supplementary angles; If parallel lines are cut by a transversal, then alternate interior angles are equal in measure 3. π¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π¦β 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¦β 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¦β π = πππ°, If parallel lines are cut by a transversal, then interior angles on the same side are supplementary MP.7 6. π¦β π = πππ°, Linear pairs form supplementary angles; If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure 7. π¦β π = ππ°, Consecutive adjacent angles on a line sum to πππ° π¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure 8. π¦β π = ππ°, If parallel lines are cut by a transversal, then corresponding angles are equal in measure Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 58 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 9. π¦β π = ππ°, Consecutive adjacent angles on a line sum to πππ° π¦β π = ππ°, If parallel lines are cut by a transversal, then corresponding angles are equal in measure MP.7 10. π¦β 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 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 β πΉπ·πΈ and β 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 β π and β π are alternate interior angles, and β π and β π are vertical angles, then β π and β π are corresponding angles. Exit Ticket (5 minutes) Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 59 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 AnglesβTransversals Exit Ticket Determine the value of each variable. π₯= π¦= π§= Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 60 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¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure MP.7 2. π¦β π = ππ°, If parallel lines are cut by a transversal, then corresponding angles are equal in measure π¦β π = ππ°, Vertical angles are equal in measure; If parallel lines are cut by a transversal, then corresponding angles are equal in measure Lesson 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 61 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 3. π¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure π¦β π = ππ°, If parallel lines are cut by a transversal, then alternate interior angles are equal in measure MP.7 4. π¦β 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 7: Date: Solve for Unknown AnglesβTransversals 4/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 62 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
