Lesson 7 M1 GEOMETRY

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Lesson 7: Solve for Unknown Angles—Transversals

Student Outcomes . 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 61

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Classwork Opening Exercise (4 minutes)

Opening Exercise

Use the diagram at the right to determine 풙 and 풚. 푨푩⃡ and 푪푫⃡ are straight lines.

풙 = ퟑퟎ 풚 = ퟓퟐ

Name a pair of vertical angles:

∠푨푶푪, ∠푫푶푩

Find the measure of ∠푩푶푭. Justify your calculation.

풎∠푩푶푭 = ퟑퟐ° 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 푨푩 and line 푪푫 in a plane (see the diagram below), a third line 푬푭 is called a transversal if it intersects 푨푩⃡ at a single point and intersects 푪푫⃡ at a single but different point. Line 푨푩 and line 푪푫 are parallel if and only if the following types of angle pairs are congruent or supplementary:

. Corresponding angles are equal in measure.

∠풂 and ∠풆 , ∠풅 and ∠풉, etc.

. Alternate interior angles are equal in measure.

∠풄 and ∠풇, ∠풅 and ∠풆

. Same-side interior angles are supplementary.

∠풄 and ∠풆, ∠풅 and ∠풇

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Examples (7 minutes)

Students try examples based on the Discussion; review and then discuss auxiliary line.

Examples

1. 2.

풎∠풂 = ퟒퟖ° 풎∠풃 = ퟏퟑퟐ°

3. 4. MP.7

풎∠풄 = ퟒퟖ° 풎∠풅 = ퟒퟖ°

5. 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 ∠풆 and ∠풇 (how?) and then add the two measures together to find the measure of ∠푾. What is the measure of ∠푾?

풎∠풆 = ퟒퟏ°, 풎∠풇 = ퟑퟓ°, 풎∠푾 = ퟕퟔ°

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 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. 풎∠풂 = ퟓퟑ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure.

풎∠풃 = ퟓퟑ° Vertical angles are equal in measure.

풎∠풄 = ퟏퟐퟕ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary.

풎∠풅 = ퟏퟒퟓ° Linear pairs form supplementary angles; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

3. MP.7 풎∠풆 = ퟓퟒ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

풎∠풇 = ퟔퟖ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then interior angles on the same side are supplementary.

풎∠품 = ퟗퟐ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then interior angles on the same side are supplementary.

풎∠풉 = ퟏퟎퟎ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

풎∠풊 = ퟏퟏퟒ° Linear pairs form supplementary angles; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

7. 풎∠풋 = ퟗퟐ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

풎∠풌 = ퟒퟐ° Consecutive adjacent angles on a line sum to ퟏퟖퟎ°.

풎∠풎 = ퟒퟔ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

MP.7 풎∠풏 = ퟖퟏ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure.

9. 풎∠풑 = ퟏퟖ° Consecutive adjacent angles on a line sum to ퟏퟖퟎ°.

풎∠풒 = ퟗퟒ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure.

10. 풎∠풓 = ퟒퟔ° 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 65

MP.7

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Relevant Vocabulary

ALTERNATE INTERIOR ANGLES: Let line 풕 be a transversal to lines 풍 and 풎 such that 풕 intersects 풍 at point 푷 and intersects 풎 at point 푸. Let 푹 be a point on line 풍 and 푺 be a point on line 풎 such that the points 푹 and 푺 lie in opposite half-planes of 풕. Then ∠푹푷푸 and ∠푷푸푺 are called alternate interior angles of the transversal 풕 with respect to line 풎 and line 풍.

CORRESPONDING ANGLES: Let line 풕 be a transversal to lines 풍 and 풎. If ∠풙 and ∠풚 are alternate interior angles and ∠풚 and ∠풛 are vertical angles, then ∠풙 and ∠풛 are corresponding angles.

Closing (1 minute) . Describe the conditions that tell us when a pair of lines cut by a transversal must be parallel.  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)

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Name Date

Lesson 7: Solving for Unknown Angles—Transversals

Exit Ticket

Determine the value of each variable.

푥 =

푦 =

푧 =

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

Exit Ticket Sample Solutions

Determine the value of each variable.

풙 = ퟐퟖ. ퟕퟓ

풚 = ퟏퟑퟓ. ퟓ

풛 = ퟒퟒ. ퟓ

Problem Set Sample Solutions

Find the unknown (labeled) angles. Give reasons for your solutions.

풎∠풂 = ퟒퟎ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

MP.7

2. 풎∠풃 = ퟒퟖ° If parallel lines are cut by a transversal, then corresponding angles are equal in measure.

풎∠풄 = ퟒퟔ° Vertical angles are equal in measure; if parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M1 GEOMETRY

3. 풎∠풅 = ퟓퟎ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

풎∠풆 = ퟓퟎ° If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.

MP.7 4.

풎∠풇 = ퟖퟐ° 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 69