Transversals with Algebra worksheet for solving angle equations using parallel lines.

Worksheet with algebraic expressions on transversals, showing parallel lines and angles for solving equations.

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Show Answer Key & Explanations Step-by-step solution for: Geometry Worksheet: Transversals and Related Angles with Algebra

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To solve the problems involving parallel lines and transversals, we need to use the properties of angles formed by a transversal intersecting parallel lines. Specifically, we will use the following angle relationships:

1. Corresponding Angles: Angles that are in the same relative position at each intersection are equal.
2. Alternate Interior Angles: Angles that lie on opposite sides of the transversal and between the two parallel lines are equal.
3. Alternate Exterior Angles: Angles that lie on opposite sides of the transversal and outside the two parallel lines are equal.
4. Consecutive Interior Angles (Same-Side Interior Angles): Angles that lie on the same side of the transversal and between the two parallel lines are supplementary (sum to 180°).

Let's solve each problem step by step.

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Problem 1

Given:
- \( 8x + 20^\circ \)
- \( 6x + 30^\circ \)

These angles are corresponding angles (or alternate interior angles), so they are equal:
\[
8x + 20 = 6x + 30
\]

Solve for \( x \):
\[
8x - 6x = 30 - 20
\]
\[
2x = 10
\]
\[
x = 5
\]

Answer for Problem 1: \( x = 5 \)

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Problem 2

Given:
- \( 13x - 12^\circ \)
- \( 4x + 20^\circ \)

These angles are corresponding angles (or alternate interior angles), so they are equal:
\[
13x - 12 = 4x + 20
\]

Solve for \( x \):
\[
13x - 4x = 20 + 12
\]
\[
9x = 32
\]
\[
x = \frac{32}{9}
\]

Answer for Problem 2: \( x = \frac{32}{9} \)

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Problem 3

Given:
- \( 5x + 7^\circ \)
- \( 6x - 23^\circ \)

These angles are consecutive interior angles, so they are supplementary:
\[
(5x + 7) + (6x - 23) = 180
\]

Solve for \( x \):
\[
5x + 6x + 7 - 23 = 180
\]
\[
11x - 16 = 180
\]
\[
11x = 196
\]
\[
x = \frac{196}{11}
\]

Answer for Problem 3: \( x = \frac{196}{11} \)

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Problem 4

Given:
- \( 15x + 10^\circ \)
- \( 10x + 35^\circ \)

These angles are corresponding angles (or alternate interior angles), so they are equal:
\[
15x + 10 = 10x + 35
\]

Solve for \( x \):
\[
15x - 10x = 35 - 10
\]
\[
5x = 25
\]
\[
x = 5
\]

Answer for Problem 4: \( x = 5 \)

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Problem 5

Given:
- \( 5x + 5^\circ \)
- \( 6x - 47^\circ \)

These angles are consecutive interior angles, so they are supplementary:
\[
(5x + 5) + (6x - 47) = 180
\]

Solve for \( x \):
\[
5x + 6x + 5 - 47 = 180
\]
\[
11x - 42 = 180
\]
\[
11x = 222
\]
\[
x = \frac{222}{11}
\]

Answer for Problem 5: \( x = \frac{222}{11} \)

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Problem 6

Given:
- \( 13x + 7^\circ \)
- \( 12x + 18^\circ \)

These angles are corresponding angles (or alternate interior angles), so they are equal:
\[
13x + 7 = 12x + 18
\]

Solve for \( x \):
\[
13x - 12x = 18 - 7
\]
\[
x = 11
\]

Answer for Problem 6: \( x = 11 \)

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Problem 7

Given:
- \( 12x - 3^\circ \)
- \( 20x - 83^\circ \)

These angles are consecutive interior angles, so they are supplementary:
\[
(12x - 3) + (20x - 83) = 180
\]

Solve for \( x \):
\[
12x + 20x - 3 - 83 = 180
\]
\[
32x - 86 = 180
\]
\[
32x = 266
\]
\[
x = \frac{266}{32} = \frac{133}{16}
\]

Answer for Problem 7: \( x = \frac{133}{16} \)

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Problem 8

Given:
- \( 5x + 1^\circ \)
- \( 7x - 15^\circ \)

These angles are corresponding angles (or alternate interior angles), so they are equal:
\[
5x + 1 = 7x - 15
\]

Solve for \( x \):
\[
5x - 7x = -15 - 1
\]
\[
-2x = -16
\]
\[
x = 8
\]

Answer for Problem 8: \( x = 8 \)

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