Geometry worksheet for calculating angles on parallel lines with clues, suitable for GCSE Foundation level.

Let's solve each of the problems on the worksheet step by step, using angle properties related to parallel lines, triangles, and angle sums.

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🔷 Question 1: Angle a

Diagram: A parallelogram with one angle marked as 72°. The opposite angle is labeled $ a^\circ $.

- In a parallelogram, opposite angles are equal.
- So, angle $ a = 72^\circ $

✔ Answer:
- Angle a: $ 72^\circ $
- Reason: Opposite angles in a parallelogram are equal.

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🔷 Question 2: Angle b

Diagram: A parallelogram with one angle $ 67^\circ $, and adjacent angle $ b^\circ $.

- In a parallelogram, adjacent angles are supplementary (sum to $ 180^\circ $).
- So, $ b + 67^\circ = 180^\circ $
- $ b = 180^\circ - 67^\circ = 113^\circ $

✔ Answer:
- Angle b: $ 113^\circ $
- Reason: Adjacent angles in a parallelogram are supplementary.

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🔷 Question 3: Angle c

Diagram: A quadrilateral with angles $ 126^\circ $, $ 108^\circ $, $ 52^\circ $, and an exterior angle $ c^\circ $ at the fourth vertex.

We need to find $ c^\circ $, which is an exterior angle at the bottom-right corner.

First, find the interior angle at that vertex:

- Sum of interior angles in a quadrilateral = $ 360^\circ $
- So, missing interior angle = $ 360^\circ - (126^\circ + 108^\circ + 52^\circ) $
- $ = 360^\circ - 286^\circ = 74^\circ $

Now, $ c^\circ $ is the exterior angle at this vertex, so:
- $ c = 180^\circ - 74^\circ = 106^\circ $

✔ Answer:
- Angle c: $ 106^\circ $
- Reason: The sum of interior angles in a quadrilateral is $ 360^\circ $. The exterior angle is supplementary to the interior angle.

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🔷 Question 4: Angle d

Diagram: Two parallel lines cut by a triangle. Angles inside the triangle: $ 68^\circ $, $ 75^\circ $, and angle $ d^\circ $ is outside the triangle, between the top line and the side.

Let’s analyze:

- The triangle has angles: $ 68^\circ $, $ 75^\circ $, and the third angle can be found:
- Sum of angles in triangle = $ 180^\circ $
- Third angle = $ 180^\circ - 68^\circ - 75^\circ = 37^\circ $

Now, angle $ d^\circ $ is formed where the top side of the triangle meets the upper parallel line.

Since the two lines are parallel, and the transversal cuts them, we can use corresponding angles or alternate angles.

But note: $ d^\circ $ is on the same side of the transversal as the $ 37^\circ $ angle, but it's outside the triangle.

Wait — actually, look closely: $ d^\circ $ is the angle between the top line and the left side of the triangle, forming a straight line with the $ 68^\circ $ angle?

No — better approach:

Actually, consider that $ d^\circ $ is an alternate angle to the angle opposite the $ 68^\circ $ and $ 75^\circ $.

Wait — let's reconsider.

The triangle has angles:
- $ 68^\circ $, $ 75^\circ $, and $ 37^\circ $

Now, the top side of the triangle forms a straight line with the top parallel line.

So, angle $ d^\circ $ is equal to the interior angle at the top of the triangle, because they are alternate interior angles? Not quite.

Wait — think differently.

Look: The triangle is sitting between two parallel lines. The left side of the triangle is a transversal.

At the top vertex, the angle is $ 68^\circ $. But angle $ d^\circ $ is outside, adjacent to $ 68^\circ $, and formed with the top parallel line.

So, $ d^\circ $ and $ 68^\circ $ are on a straight line, meaning they form a linear pair.

Wait — no! They are not on a straight line unless the top edge is extended.

Wait — the diagram shows the top line extending beyond the triangle, and $ d^\circ $ is between the extension and the side of the triangle.

So, $ d^\circ $ is an exterior angle at the top vertex.

But more clearly: Since the top line is parallel to the bottom line, and the side of the triangle is a transversal, then:

- The angle $ d^\circ $ and the interior angle at the top of the triangle ($ 68^\circ $) are alternate interior angles?

Wait — no. Let’s label carefully.

Actually, the interior angle at the top of the triangle is $ 68^\circ $. The angle $ d^\circ $ is between the top line and the side, and since the top line is parallel to the base, and the side is a transversal, then:

- $ d^\circ $ and $ 68^\circ $ are co-interior angles? No.

Wait — better idea: Use triangle angles.

The triangle has angles: $ 68^\circ $, $ 75^\circ $, and unknown at the bottom.

Wait — no, the $ 68^\circ $ and $ 75^\circ $ are two of the three angles of the triangle.

So third angle = $ 180 - 68 - 75 = 37^\circ $

Now, this $ 37^\circ $ is at the bottom-left of the triangle.

Now, the top side of the triangle is part of a transversal across two parallel lines.

So, the angle $ d^\circ $ is alternate to the angle at the bottom-left of the triangle, which is $ 37^\circ $, because the lines are parallel.

Yes!

- $ d^\circ $ and $ 37^\circ $ are alternate interior angles, so they are equal.

✔ Answer:
- Angle d: $ 37^\circ $
- Reason: Alternate interior angles are equal when lines are parallel.

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🔷 Question 5: Angle e and f

Diagram: Two intersecting lines, with a triangle. One angle is $ 81^\circ $, another $ 39^\circ $, and $ f^\circ $ is in the triangle, $ e^\circ $ is an angle formed by intersection.

Let’s break it down.

We have a triangle with angles:
- $ 39^\circ $
- $ f^\circ $
- and another angle adjacent to $ 81^\circ $

Wait — there's an angle of $ 81^\circ $ at the top, and $ f^\circ $ is the angle inside the triangle at that vertex.

But wait — the $ 81^\circ $ is outside the triangle, and $ f^\circ $ is inside.

They are adjacent angles on a straight line?

Yes — the line from the top vertex goes through the triangle and continues.

So, $ f^\circ $ and $ 81^\circ $ are adjacent angles forming a straight line?

No — look: the $ 81^\circ $ is on the extension of the side, so:

- $ f^\circ $ and $ 81^\circ $ are supplementary if they lie on a straight line.

But the triangle has a $ 39^\circ $ angle at the bottom-left.

Now, the third angle of the triangle is $ f^\circ $, and we know two angles: $ 39^\circ $, and $ f^\circ $, and the third angle?

Wait — no. The triangle has only three angles.

Let’s list what we know:

- One angle of the triangle is $ 39^\circ $
- Another angle is $ f^\circ $
- The third angle is adjacent to $ 81^\circ $

But $ 81^\circ $ is outside the triangle, so the interior angle at the top is $ 180^\circ - 81^\circ = 99^\circ $? No.

Wait — $ 81^\circ $ is on the straight line, so the interior angle of the triangle at the top is $ 180^\circ - 81^\circ = 99^\circ $?

Wait — no. If $ 81^\circ $ is outside, then the interior angle is $ 180^\circ - 81^\circ = 99^\circ $? Only if they are on a straight line.

Yes — if the $ 81^\circ $ is adjacent to the triangle’s angle, then the triangle’s angle is $ 180^\circ - 81^\circ = 99^\circ $

But that would make the triangle have angles: $ 39^\circ $, $ 99^\circ $, and $ f^\circ $

Sum must be $ 180^\circ $, so:
- $ f = 180 - 39 - 99 = 42^\circ $

So $ f = 42^\circ $

Now, angle $ e^\circ $ is formed at the intersection of two lines. It appears to be vertically opposite to the angle at the top of the triangle.

Wait — the diagram shows two lines crossing: one is the side of the triangle, the other is a diagonal.

But $ e^\circ $ is at the intersection, and looks like it's vertically opposite to the angle at the top of the triangle.

But the angle at the top of the triangle is $ 99^\circ $? No — earlier we said the interior angle is $ 99^\circ $, but $ e^\circ $ is not that.

Wait — actually, $ e^\circ $ is the angle between the two transversals, and it's equal to the angle in the triangle at the bottom-left? Or perhaps corresponding.

Wait — look again.

There’s a triangle with:
- Bottom-left angle: $ 39^\circ $
- Top angle: $ f^\circ $
- And the angle adjacent to $ 81^\circ $ is $ 180^\circ - 81^\circ = 99^\circ $

Wait — but $ f^\circ $ is not the top angle. Wait — the diagram labels $ f^\circ $ as the angle in the triangle at the top.

But $ 81^\circ $ is outside, so the interior angle is $ 180^\circ - 81^\circ = 99^\circ $, so $ f = 99^\circ $? That contradicts earlier.

Wait — I think I made a mistake.

Let me re-read: The angle $ f^\circ $ is inside the triangle, and $ 81^\circ $ is outside, adjacent to it.

So, $ f^\circ $ and $ 81^\circ $ are adjacent angles on a straight line, so:

- $ f + 81^\circ = 180^\circ $
- $ f = 180^\circ - 81^\circ = 99^\circ $

Now, the triangle has:
- One angle: $ 39^\circ $
- One angle: $ f = 99^\circ $
- So third angle = $ 180 - 39 - 99 = 42^\circ $

Now, angle $ e^\circ $ is at the intersection point of two lines. One line is the side of the triangle, the other is a transversal.

Notice that $ e^\circ $ is vertically opposite to the $ 42^\circ $ angle (the third angle of the triangle), because they are formed by intersecting lines.

So, $ e = 42^\circ $

✔ Answers:
- Angle e: $ 42^\circ $
- Reason: Vertically opposite angles are equal.
- Angle f: $ 99^\circ $
- Reason: Angles on a straight line sum to $ 180^\circ $; $ f = 180^\circ - 81^\circ $.

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🔷 Question 6: Angle g

Diagram: A triangle with angles $ 21^\circ $, $ 74^\circ $, and an exterior angle $ g^\circ $ at the top.

So, first find the third interior angle of the triangle:

- $ 180^\circ - 21^\circ - 74^\circ = 85^\circ $

Now, angle $ g^\circ $ is the exterior angle at the top vertex.

An exterior angle is equal to the sum of the two opposite interior angles.

So,
- $ g = 21^\circ + 74^\circ = 95^\circ $

Alternatively, $ g = 180^\circ - 85^\circ = 95^\circ $

✔ Answer:
- Angle g: $ 95^\circ $
- Reason: Exterior angle of a triangle equals the sum of the two opposite interior angles.

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✔ Final Answers Summary:

| Angle | Value | Reason |
|------|-------|--------|
| a | $ 72^\circ $ | Opposite angles in a parallelogram are equal |
| b | $ 113^\circ $ | Adjacent angles in a parallelogram are supplementary |
| c | $ 106^\circ $ | Interior angle sum of quadrilateral is $ 360^\circ $, exterior angle is supplementary |
| d | $ 37^\circ $ | Alternate interior angles are equal (parallel lines) |
| e | $ 42^\circ $ | Vertically opposite angles are equal |
| f | $ 99^\circ $ | Angles on a straight line sum to $ 180^\circ $ |
| g | $ 95^\circ $ | Exterior angle equals sum of opposite interior angles |

Let me know if you'd like a visual explanation or diagram annotations!