Educational worksheet for GCSE Foundation level on angles formed by parallel lines, including diagrams and reasoning prompts.

Here is the complete solution for Angles on Parallel Lines (A) With Clues, broken down by section and diagram, with clear reasoning based on the provided clues.

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## ✔ Section A: State the missing angle and give a reason

1. Angle `a°`

- Diagram: Two parallel lines cut by a transversal, forming an “F” shape. The top angle is 48°, and `a°` is the corresponding angle below.
- Answer:
Angle a: 48°
Reason: It is a corresponding angle and therefore equal.

> ✔ *Corresponding angles are equal when two parallel lines are cut by a transversal.*

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2. Angle `b°`

- Diagram: Two parallel lines cut by a transversal. Top left angle is 52°, and `b°` is on the opposite side of the transversal but inside the parallels — forming a “Z” shape.
- Answer:
Angle b: 52°
Reason: It is an alternate angle and therefore equal.

> ✔ *Alternate angles (also called Z-angles) are equal when two parallel lines are cut by a transversal.*

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3. Angle `c°`

- Diagram: Two parallel lines cut by a transversal. The top left angle is 65°, and `c°` is on the same side of the transversal, inside the parallels — forming a “C” shape.
- Answer:
Angle c: 115°
Reason: It is an interior angle and therefore adds to 180°.

> ✔ *Co-interior angles (or consecutive interior angles) add up to 180° when two parallel lines are cut by a transversal.*
> So, `c = 180° - 65° = 115°`

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4. Angle `d°`

- Diagram: Two parallel lines cut by a transversal. Top right angle is 71°, and `d°` is vertically opposite to the corresponding angle.
- Answer:
Angle d: 71°
Reason: It is a corresponding angle and therefore equal. *(Also acceptable: vertically opposite angle and therefore equal — since it’s vertically opposite to the corresponding angle.)*

> ✔ *Corresponding angles are equal. Alternatively, if you trace it: the angle corresponding to 71° is equal, and `d°` is vertically opposite that — so also 71°.*

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## ✔ Section B: Calculate the missing angle and give a reason

1. Angle `a°`

- Diagram: Two parallel lines cut by a transversal. On the upper line, the angle above the transversal is 108°. `a°` is on the lower line, on the same side of the transversal — forming a “C” shape.
- Answer:
Angle a: 72°
Reason: It is an interior angle and therefore adds to 180°.

> ✔ *Co-interior angles sum to 180°.*
> So, `a = 180° - 108° = 72°`

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2. Angle `b°`

- Diagram: Two parallel lines cut by a transversal. Top right angle is 43°, and `b°` is on the lower line, on the opposite side of the transversal — forming a “Z” shape.
- Answer:
Angle b: 43°
Reason: It is an alternate angle and therefore equal.

> ✔ *Alternate angles are equal.*

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3. Angle `c°`

- Diagram: Two parallel lines cut by a transversal. Upper left angle is 114°, and `c°` is on the lower line, on the same side of the transversal — forming a “C” shape.
- Answer:
Angle c: 66°
Reason: It is an interior angle and therefore adds to 180°.

> ✔ *Co-interior angles sum to 180°.*
> So, `c = 180° - 114° = 66°`

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4. Angle `d°`

- Diagram: Two parallel lines cut by a transversal. Lower angle is 79°, and `d°` is vertically opposite to the corresponding angle on the top line.
- Answer:
Angle d: 79°
Reason: It is a corresponding angle and therefore equal. *(Also acceptable: vertically opposite angle and therefore equal — since `d°` is vertically opposite to the corresponding angle.)*

> ✔ *Corresponding angles are equal. Since the 79° angle corresponds to the angle directly above it, and `d°` is vertically opposite that, it’s also 79°.*

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## 📋 Final Answers Summary:

Section A:

| Angle | Value | Reason |
|-------|--------|--------|
| a | 48° | Corresponding angle and therefore equal |
| b | 52° | Alternate angle and therefore equal |
| c | 115° | Interior angle and therefore adds to 180° |
| d | 71° | Corresponding angle and therefore equal |

Section B:

| Angle | Value | Reason |
|-------|--------|--------|
| a | 72° | Interior angle and therefore adds to 180° |
| b | 43° | Alternate angle and therefore equal |
| c | 66° | Interior angle and therefore adds to 180° |
| d | 79° | Corresponding angle and therefore equal |

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✔ All answers use the given clue options exactly as instructed. This worksheet reinforces key GCSE Foundation geometry concepts: corresponding, alternate, co-interior, and vertically opposite angles on parallel lines.

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