Linear Pair worksheet with angle equations to solve for x.

To solve the problem of finding the measure of each angle in the given diagrams, we need to use basic geometric principles such as the properties of angles formed by parallel lines and transversals, vertical angles, supplementary angles, and complementary angles. Let's analyze each diagram step by step.

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Diagram A

- The diagram shows two parallel lines cut by a transversal.
- One angle is labeled as \( 50^\circ \).
- We need to find the measures of the other angles.

#### Step-by-Step Solution:
1. Identify Corresponding Angles:
- The angle corresponding to the \( 50^\circ \) angle is also \( 50^\circ \) because corresponding angles are equal when the lines are parallel.

2. Identify Alternate Interior Angles:
- The alternate interior angle to the \( 50^\circ \) angle is also \( 50^\circ \).

3. Identify Same-Side Interior Angles:
- The same-side interior angle adjacent to the \( 50^\circ \) angle is supplementary to it. Therefore, it is \( 180^\circ - 50^\circ = 130^\circ \).

4. Identify Vertical Angles:
- The vertical angle opposite the \( 50^\circ \) angle is also \( 50^\circ \).
- The vertical angle opposite the \( 130^\circ \) angle is also \( 130^\circ \).

#### Final Measures:
- The angles are: \( 50^\circ, 50^\circ, 130^\circ, 130^\circ \).

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Diagram B

- The diagram shows two intersecting lines forming vertical angles.
- One angle is labeled as \( 70^\circ \).
- We need to find the measures of the other angles.

#### Step-by-Step Solution:
1. Identify Vertical Angles:
- The vertical angle opposite the \( 70^\circ \) angle is also \( 70^\circ \).

2. Identify Supplementary Angles:
- The angles adjacent to the \( 70^\circ \) angle are supplementary to it. Therefore, each of these angles is \( 180^\circ - 70^\circ = 110^\circ \).

#### Final Measures:
- The angles are: \( 70^\circ, 70^\circ, 110^\circ, 110^\circ \).

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Diagram C

- The diagram shows a triangle with one angle labeled as \( 60^\circ \) and another angle labeled as \( 40^\circ \).
- We need to find the measure of the third angle.

#### Step-by-Step Solution:
1. Sum of Angles in a Triangle:
- The sum of the interior angles in any triangle is \( 180^\circ \).

2. Calculate the Third Angle:
- Let the third angle be \( x \).
- Using the angle sum property: \( 60^\circ + 40^\circ + x = 180^\circ \).
- Simplify: \( 100^\circ + x = 180^\circ \).
- Solve for \( x \): \( x = 180^\circ - 100^\circ = 80^\circ \).

#### Final Measure:
- The third angle is \( 80^\circ \).

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Diagram D

- The diagram shows a right triangle with one angle labeled as \( 30^\circ \).
- We need to find the measure of the other non-right angle.

#### Step-by-Step Solution:
1. Sum of Angles in a Triangle:
- The sum of the interior angles in any triangle is \( 180^\circ \).

2. Right Angle:
- One angle in the triangle is \( 90^\circ \).

3. Calculate the Third Angle:
- Let the unknown angle be \( x \).
- Using the angle sum property: \( 30^\circ + 90^\circ + x = 180^\circ \).
- Simplify: \( 120^\circ + x = 180^\circ \).
- Solve for \( x \): \( x = 180^\circ - 120^\circ = 60^\circ \).

#### Final Measure:
- The other non-right angle is \( 60^\circ \).

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Diagram E

- The diagram shows a straight line with one angle labeled as \( 120^\circ \).
- We need to find the measure of the adjacent angle.

#### Step-by-Step Solution:
1. Supplementary Angles:
- Angles on a straight line are supplementary, meaning their sum is \( 180^\circ \).

2. Calculate the Adjacent Angle:
- Let the adjacent angle be \( x \).
- Using the supplementary property: \( 120^\circ + x = 180^\circ \).
- Solve for \( x \): \( x = 180^\circ - 120^\circ = 60^\circ \).

#### Final Measure:
- The adjacent angle is \( 60^\circ \).

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Final Answers:

\[
\boxed{
\text{A: } 50^\circ, 50^\circ, 130^\circ, 130^\circ;
\text{B: } 70^\circ, 70^\circ, 110^\circ, 110^\circ;
\text{C: } 80^\circ;
\text{D: } 60^\circ;
\text{E: } 60^\circ
}
\]