Calculating Angles on Parallel Lines with Transversals (A) (with ... - Free Printable

Let's solve each angle problem step by step using the given clues and geometric rules for angles on parallel lines.

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Section A

#### 1. Angle a:
- Given: 48°
- The two lines are parallel, and the transversal forms an alternate interior angle with angle $ a $.
- Since the 48° angle and angle $ a $ are alternate interior angles, they are equal.

✔ Angle a: 48°
✔ Reason: It is an alternate angle and therefore equal.

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#### 2. Angle b:
- Given: 52°
- The angle $ b $ is opposite to the 52° angle formed at the intersection of the transversal and the top line.
- These two angles are vertically opposite angles (formed when two lines cross), so they are equal.

✔ Angle b: 52°
✔ Reason: It is a vertically opposite angle and therefore equal.

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#### 3. Angle c:
- Given: 65°
- The two lines are parallel, and the angle $ c $ is adjacent to the 65° angle along a straight line.
- Since angles on a straight line add up to 180°, we can calculate:
$$
c = 180^\circ - 65^\circ = 115^\circ
$$

✔ Angle c: 115°
✔ Reason: Angles on a straight line add up to 180°.

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#### 4. Angle d:
- Given: 71°
- The 71° angle and angle $ d $ are corresponding angles (same position relative to the transversal and parallel lines).
- Corresponding angles are equal when lines are parallel.

✔ Angle d: 71°
✔ Reason: It is a corresponding angle and therefore equal.

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Section B

#### 5. Angle a:
- Given: 108°
- The 108° angle and angle $ a $ are interior angles on the same side of the transversal between parallel lines.
- Interior angles on the same side of the transversal add up to 180°.
$$
a = 180^\circ - 108^\circ = 72^\circ
$$

✔ Angle a: 72°
✔ Reason: It is an interior angle and therefore adds to 180°.

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#### 6. Angle b:
- Given: 43°
- The 43° angle and angle $ b $ are alternate interior angles.
- Alternate interior angles are equal when lines are parallel.

✔ Angle b: 43°
✔ Reason: It is an alternate angle and therefore equal.

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#### 7. Angle c:
- Given: 114°
- The 114° angle and angle $ c $ are corresponding angles (they are in the same relative position on the parallel lines).
- Therefore, they are equal.

✔ Angle c: 114°
✔ Reason: It is a corresponding angle and therefore equal.

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#### 8. Angle d:
- Given: 79°
- The 79° angle and angle $ d $ are vertically opposite angles (they are opposite each other at the intersection).
- Vertically opposite angles are equal.

✔ Angle d: 79°
✔ Reason: It is a vertically opposite angle and therefore equal.

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✔ Final Answers:

| Angle | Value | Reason |
|-------|--------|--------|
| a | 48° | It is an alternate angle and therefore equal. |
| b | 52° | It is a vertically opposite angle and therefore equal. |
| c | 115° | Angles on a straight line add up to 180°. |
| d | 71° | It is a corresponding angle and therefore equal. |
| a | 72° | It is an interior angle and therefore adds to 180°. |
| b | 43° | It is an alternate angle and therefore equal. |
| c | 114° | It is a corresponding angle and therefore equal. |
| d | 79° | It is a vertically opposite angle and therefore equal. |

> 🔍 Note: The letter "a", "b", etc., are reused in both sections. Make sure to distinguish them based on their diagrams.

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