Geometry worksheet for calculating missing angles on parallel lines, featuring six problems with diagrams and spaces for answers and reasoning.

Let's solve each of the problems on the worksheet "Angles on Parallel Lines (B)", calculating the missing angles and providing a clear reason for each answer.

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1. Angle a:

Diagram: A trapezium with two parallel lines (indicated by arrows), and one angle given as 72°, and we need to find angle $ a^\circ $ at the bottom right.

- The 72° angle and angle $ a $ are co-interior (same-side interior) angles formed by a transversal cutting two parallel lines.
- Co-interior angles add up to 180°.

$$
a = 180^\circ - 72^\circ = 108^\circ
$$

✔ Angle a: 108°
✔ Reason: Co-interior angles on parallel lines add up to 180°.

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2. Angle b:

Diagram: A parallelogram (opposite sides parallel), with one angle given as 67°, and we need to find angle $ b^\circ $ at the top left.

- In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°).
- Since 67° is adjacent to angle $ b $, they are co-interior angles.

$$
b = 180^\circ - 67^\circ = 113^\circ
$$

✔ Angle b: 113°
✔ Reason: Adjacent angles in a parallelogram are supplementary (sum to 180°).

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3. Angle c:

Diagram: A quadrilateral with three angles labeled: 126°, 108°, and 52°, and angle $ c^\circ $ is an exterior angle at the bottom right.

We can find the interior angle first using the sum of interior angles in a quadrilateral:

Sum of interior angles in a quadrilateral = $ 360^\circ $

Add known interior angles:
$$
126^\circ + 108^\circ + 52^\circ = 286^\circ
$$

So, the fourth interior angle is:
$$
360^\circ - 286^\circ = 74^\circ
$$

Now, angle $ c $ is the exterior angle adjacent to this 74° interior angle.

Exterior angle = $ 180^\circ - 74^\circ = 106^\circ $

✔ Angle c: 106°
✔ Reason: Exterior angle = 180° – interior angle; sum of angles in a quadrilateral is 360°.

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4. Angle d:

Diagram: A triangle between two parallel lines, with angles 68° and 75° inside the triangle, and angle $ d^\circ $ is at the top left, formed between the upper parallel line and the side of the triangle.

Let’s analyze:

- The triangle has internal angles: 68° and 75°, so the third angle (at the top vertex) is:
$$
180^\circ - 68^\circ - 75^\circ = 37^\circ
$$

Now, angle $ d $ is equal to this 37° angle because it is an alternate interior angle formed by the transversal crossing the parallel lines.

So, $ d = 37^\circ $

✔ Angle d: 37°
✔ Reason: Alternate interior angles are equal when lines are parallel.

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5. Angle e and f:

Diagram: Two intersecting lines forming a triangle, with angles 81°, 39°, and unknowns $ e $ and $ f $. Also, there's a transversal across parallel lines.

First, let's find angle $ f $:

- In the triangle, two angles are 81° and 39°.
- So, the third angle $ f $ is:
$$
f = 180^\circ - 81^\circ - 39^\circ = 60^\circ
$$

✔ Angle f: 60°
✔ Reason: Sum of angles in a triangle is 180°.

Now, angle $ e $: It is formed where the transversal crosses the parallel lines. Note that angle $ e $ is vertically opposite to the angle adjacent to $ f $.

Wait — actually, look at the diagram: the 81° angle is at the top, and the line extends beyond, forming angle $ e $.

But notice: the angle adjacent to $ f $ (which is 60°) and angle $ e $ are corresponding angles or alternate angles?

Actually, better approach:

The angle at the intersection of the transversal and the lower line forms a straight line with angle $ f $. But wait — angle $ e $ is on the opposite side of the transversal, and since the lines are parallel, and angle $ e $ is vertically opposite to the angle formed at the intersection.

Wait — simpler:

Angle $ e $ is vertically opposite to the angle adjacent to the 60° angle? Let's think differently.

Actually, the triangle has angles 81°, 39°, and 60° (we found $ f = 60^\circ $). Now, the line extending from the 81° angle forms angle $ e $ with the transversal.

But observe: angle $ e $ and the 81° angle are vertically opposite? No.

Wait — the 81° angle is between the upper parallel line and the diagonal line. Then the extension of that diagonal creates angle $ e $ with the lower parallel line.

Since the lines are parallel, and the diagonal is a transversal, then:

- The 81° angle and angle $ e $ are corresponding angles?

No — they are not on the same side.

Alternatively, note that angle $ e $ is equal to the 81° angle because they are alternate interior angles?

Wait — let’s re-express.

The key is: angle $ e $ is formed between the transversal and the lower parallel line, and it is on the same side as the 81° angle, but opposite.

Actually, angle $ e $ and the 81° angle are corresponding angles if the transversal cuts both parallel lines.

Yes! They are corresponding angles.

Therefore:
$$
e = 81^\circ
$$

✔ Angle e: 81°
✔ Reason: Corresponding angles are equal when lines are parallel.

And earlier:

✔ Angle f: 60°
✔ Reason: Sum of angles in a triangle is 180°.

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6. Angle g:

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

We can use the exterior angle theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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

✔ Angle g: 95°
✔ Reason: Exterior angle of a triangle equals the sum of the two opposite interior angles.

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✔ Final Answers:

| Angle | Value | Reason |
|-------|--------|--------|
| a | 108° | Co-interior angles add to 180° |
| b | 113° | Adjacent angles in a parallelogram are supplementary |
| c | 106° | Exterior angle = 180° – interior angle; sum of quad = 360° |
| d | 37° | Alternate interior angles are equal |
| e | 81° | Corresponding angles are equal |
| f | 60° | Sum of angles in a triangle is 180° |
| g | 95° | Exterior angle = sum of opposite interior angles |

Let me know if you'd like a visual explanation or help with any specific diagram!