Unit 2 Day 4: Solving Problems with Angle Relationships

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Unit 2: Lines, Angles, and Parallelism

Day 4: Solving Problems

Today's Objective

Warm-Up: Day 3 Review

If \(m\angle4 = (x+20)^\circ\) and \(m\angle5 = (2x)^\circ\), what value of x makes the lines parallel?

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Back to the Original Logic

YESTERDAY'S LOGIC (THE CONVERSE)

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

TODAY'S LOGIC

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

Today, we start by KNOWING the lines are parallel. This lets us make conclusions about the angles.

Angle Relationships with Parallel Lines

If two parallel lines are cut by a transversal, the angle pairs are either CONGRUENT or SUPPLEMENTARY.

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Congruent (≅)

  • Corresponding
  • Alternate Interior
  • Alternate Exterior
  • Vertical

Supplementary (= 180°)

  • Same-Side Interior
  • Linear Pair

I Do: Finding Angle Measures

l m 2 125°

GIVEN: l || m and \(m\angle2 = 125^\circ\).

FIND: The measures of all other angles.

\(m\angle6 = 125^\circ\) (Corresponding)

\(m\angle7 = 125^\circ\) (Alternate Exterior)

\(m\angle3 = 125^\circ\) (Vertical)

All others are \(180-125 = 55^\circ\).

I Do: Solving for x (Congruent Angles)

GIVEN: l || m.

SETUP: ∠4 and ∠5 are alternate interior angles. Since the lines are parallel, the angles are congruent.

\(x+20 = 2x-15\)

\(35 = x\)

l m 4 (x+20)° 5 (2x-15)°

I Do: Solving for x (Supplementary Angles)

GIVEN: l || m.

SETUP: ∠3 and ∠5 are same-side interior angles. Since the lines are parallel, the angles are supplementary.

\((x+50) + (2x-20) = 180\)

\(3x + 30 = 180\)

\(3x = 150 \implies x = 50\)

l m 3 (x+50)° 5 (2x-20)°

We Do: Let's Try Together

Given l || m, find the missing value.

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1. If \(m\angle2 = 60^\circ\), find \(m\angle7\). Why?

2. If \(m\angle3 = 110^\circ\), find \(m\angle5\). Why?

You Do: Practice Time!

Now, open your interactive notes to Day 4 and complete the "You Do" section.

Independent Practice

Given l || m, find the value of x.

Green Level

\(m\angle1 = 130^\circ\)

\(m\angle8 = x^\circ\)

Yellow Level

\(m\angle2 = (5x - 10)^\circ\)

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

Red Level

\(m\angle4 = (x + 25)^\circ\)

\(m\angle6 = (4x - 15)^\circ\)

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Exit Ticket

Given l || m, find the measure of angle 4.

\(m\angle2 = (3x+15)^\circ\)

\(m\angle5 = (5x+5)^\circ\)

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Unit 2: Lines, Angles, and Parallelism

Day 4: Solving Problems

Warm-Up: Day 3 Review

If \(m\angle4 = (x+20)^\circ\) and \(m\angle5 = (2x)^\circ\), what value of x makes the lines parallel?

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Angle Relationships with Parallel Lines

If two parallel lines are cut by a transversal, the angle pairs are either CONGRUENT or SUPPLEMENTARY.

12 43 56 87

Congruent (≅)

  • Corresponding
  • Alternate Interior
  • Alternate Exterior
  • Vertical

Supplementary (= 180°)

  • Same-Side Interior
  • Linear Pair

Reference Diagram for Practice

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We Do: Let's Try Together

1. If \(m\angle2 = 60^\circ\), find \(m\angle7\). . Why?

2. If \(m\angle3 = 110^\circ\), find \(m\angle5\). . Why?

Independent Practice

Given l || m, find the value of x.

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Green: \(m\angle1 = 130^\circ\), \(m\angle8 = x^\circ\). x =

Yellow: \(m\angle2 = (5x - 10)^\circ\), \(m\angle6 = (3x + 20)^\circ\). x =

Red: \(m\angle4 = (x + 25)^\circ\), \(m\angle6 = (4x - 15)^\circ\). x =

Exit Ticket

Given l || m, first find x, then find the measure of angle 4.

\(m\angle2 = (3x+15)^\circ\), \(m\angle5 = (5x+5)^\circ\)

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x =

m∠4 =