Unit 2 Day 3: Proving Lines Parallel

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Unit 2: Lines, Angles, and Parallelism

Day 3: Proving Lines Parallel

Today's Objective

Warm-Up: Day 2 Review

What is the name of the angle pair relationship between ∠3 and ∠6?

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The Big Shift: Reversing Our Logic

YESTERDAY'S LOGIC

IF lines are parallel, THEN alternate interior angles are congruent.

TODAY'S LOGIC (THE CONVERSE)

IF alternate interior angles are congruent, THEN lines are parallel.

Vocabulary: The Converse

The converse of a theorem is created by switching the "if" and "then" parts.

Statement: If it is raining, then the ground is wet.

Converse: If the ground is wet, then it is raining.

(Note: In real life, the converse isn't always true! But in our geometry proofs today, they are!)

I Do: Converse of Alt. Interior Angles

GIVEN: \(m\angle3 = 80^\circ\) and \(m\angle6 = 80^\circ\).

OBSERVE: ∠3 and ∠6 are alternate interior angles, and they are congruent.

CONCLUDE: Line l is parallel to line m.

l m 3 80° 6 80°

I Do: Converse of Corresponding Angles

GIVEN: \(m\angle1 = 115^\circ\) and \(m\angle5 = 115^\circ\).

OBSERVE: ∠1 and ∠5 are corresponding angles, and they are congruent.

CONCLUDE: Line l is parallel to line m.

l m 1 115° 5 115°

I Do: Converse of Same-Side Interior

GIVEN: \(m\angle3 = 100^\circ\) and \(m\angle5 = 80^\circ\).

OBSERVE: ∠3 and ∠5 are same-side interior angles, and they are supplementary (100 + 80 = 180).

CONCLUDE: Line l is parallel to line m.

l m 3 100° 5 80°

We Do: Let's Try Together

What theorem proves l || m for each case?

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1. If you are given that ∠4 ≅ ∠5?

2. If you are given m∠3 + m∠5 = 180°?

You Do: Practice Time!

Now, open your interactive notes to Day 3 and complete the "You Do" section.

Independent Practice

Find the value of x that makes lines l and m parallel.

Green Level

\(m\angle2 = 75^\circ\)

\(m\angle6 = (5x)^\circ\)

Yellow Level

\(m\angle3 = (2x + 10)^\circ\)

\(m\angle6 = (3x - 20)^\circ\)

Red Level

\(m\angle4 = (4x - 10)^\circ\)

\(m\angle5 = (x + 15)^\circ\)

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Exit Ticket

Given \(m\angle2 = 120^\circ\), what must \(m\angle7\) be to prove the lines are parallel?

Which theorem or postulate justifies your answer?

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Unit 2: Lines, Angles, and Parallelism

Day 3: Proving Lines Parallel

Warm-Up: Day 2 Review

What is the name of the angle pair relationship between ∠3 and ∠6?

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Vocabulary: The Converse

The converse of a theorem is created by the "if" and "then" parts.

The Converse Theorems & Postulates

Converse of Corresponding Angles: IF corresponding angles are congruent, THEN lines are .

Converse of Alternate Interior Angles: IF alternate interior angles are congruent, THEN lines are .

Converse of Same-Side Interior Angles: IF same-side interior angles are , THEN lines are parallel.

Reference Diagram for Practice

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We Do: Let's Try Together

1. If you are given ∠4 ≅ ∠5, which theorem proves l || m?

2. If you are given m∠3 + m∠5 = 180°, which theorem proves l || m?

Independent Practice

Find the value of x that makes lines l and m parallel.

Green: \(m\angle2 = 75^\circ\), \(m\angle6 = (5x)^\circ\). x =

Yellow: \(m\angle3 = (2x + 10)^\circ\), \(m\angle6 = (3x - 20)^\circ\). x =

Red: \(m\angle4 = (4x - 10)^\circ\), \(m\angle5 = (x + 15)^\circ\). x =

Exit Ticket

Given \(m\angle2 = 120^\circ\), what must \(m\angle7\) be to prove l || m?

Which theorem or postulate justifies your answer?