Geometry worksheet for calculating missing angles on parallel lines with provided clues.

Let's solve each of the problems on the worksheet "Angles on Parallel Lines (B) With Clues" step by step, using geometric principles related to angles on parallel lines.

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🔷 Question 1: Angle a

- Given: A parallelogram with one angle = 72°.
- The angle marked a° is opposite to the 72° angle.

#### ✔ Solution:
In a parallelogram, opposite angles are equal.

So:
> Angle a = 72°

✔ Reason: Opposite angles in a parallelogram are equal.

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🔷 Question 2: Angle b

- Given: A parallelogram with one angle = 67°, and we're to find angle b°, which is adjacent to it.

#### ✔ Solution:
In a parallelogram, adjacent angles are supplementary (add up to 180°).

So:
> Angle b = 180° – 67° = 113°

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

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🔷 Question 3: Angle c

- Given: A quadrilateral with angles: 126°, 108°, 52°, and an exterior angle c° at the fourth vertex.

We need to find angle c, which is an exterior angle.

First, find the interior angle at that vertex.

Sum of interior angles in a quadrilateral = 360°

So:
> Interior angle at vertex C = 360° – (126° + 108° + 52°)
> = 360° – 286° = 74°

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

Exterior angle = 180° – interior angle
> So: c = 180° – 74° = 106°

✔ Reason: Exterior angle = 180° minus the interior angle; sum of angles in a quadrilateral is 360°.

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🔷 Question 4: Angle d

- Given: A triangle with two angles: 68° and 75°. The third angle is formed between two parallel lines, and we’re to find angle d° at the top.

But wait — look carefully: there are two parallel lines with a triangle drawn between them. The angle d° is outside the triangle, but it’s part of a straight line with the triangle’s top angle.

First, find the third angle of the triangle:

Sum of angles in triangle = 180°
So:
> Third angle = 180° – (68° + 75°) = 180° – 143° = 37°

Now, angle d is on a straight line with this 37° angle, so they are supplementary.

> d = 180° – 37° = 143°

✔ Reason: Angles on a straight line add to 180°; sum of angles in a triangle is 180°.

Alternatively, you might recognize that d is a corresponding or alternate angle, but here it’s more direct via triangle and straight line.

Wait — actually, let's reconsider: d is at the top vertex, and the two sides go along parallel lines. The triangle has angles 68° and 75°, so the third internal angle is 37°, and d is the same as the 37° angle? No — d is outside the triangle, adjacent to the 37° angle.

So yes, d = 180° – 37° = 143°

✔ Answer: 143°, Reason: Angles on a straight line add to 180°.

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🔷 Question 5: Angle e and f

This diagram shows two intersecting lines forming angles with parallel lines.

Given:
- One angle = 81°
- Another angle = 39°
- We need to find e° and f°

Let’s analyze:

There’s a triangle with angles 39° and f°, and another angle at the top of the triangle is 81°.

Wait — actually, the 81° angle is at the intersection point, and the lines are parallel.

Let’s break it down:

- The 81° angle is between a transversal and a parallel line.
- The 39° is inside a triangle.
- We have two parallel lines crossed by a transversal.

Let’s consider the triangle:
- It has one angle = 39°
- Another angle is f°
- The third angle is adjacent to the 81° angle.

But note: the 81° angle is corresponding or alternate to some other angle?

Actually, observe: the 81° angle and f° are on a straight line across from each other? Wait — no.

Better approach:

Look at the top vertex: the 81° angle is between two lines — one is a transversal, the other is a side of the triangle.

But also, the triangle has angles: 39°, f, and the angle at the top.

But the top angle of the triangle is adjacent to the 81° angle.

So:
The angle inside the triangle at the top = 180° – 81° = 99°

Now, in the triangle:
- Angles: 39°, 99°, and f°
- Sum = 180°

So:
> f = 180° – 39° – 99° = 42°

Now, angle e is the vertically opposite angle to the 81° angle? Or is it alternate?

Wait — look: e° is at the intersection of the transversal and the lower parallel line.

And 81° is on the upper line.

Since the lines are parallel, and the transversal crosses both, then:

> e° = 81° because they are corresponding angles.

✔ So:
- Angle e = 81°, Reason: Corresponding angles are equal when lines are parallel.
- Angle f = 42°, Reason: Angles in a triangle add to 180°.

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🔷 Question 6: Angle g

- Given: A triangle with angles 21° and 74°, and an exterior angle g° at the top vertex.

We need to find g°, which is an exterior angle.

First, find the interior angle at the top vertex.

Sum of angles in triangle = 180°
So:
> Interior angle at top = 180° – (21° + 74°) = 180° – 95° = 85°

Then, g° is the exterior angle adjacent to this 85° angle.

So:
> g = 180° – 85° = 95°

✔ Reason: Angles on a straight line sum to 180°.

Alternatively, use the exterior angle theorem: exterior angle = sum of opposite interior angles.

So:
> g = 21° + 74° = 95°

✔ That's faster!

So:
> Angle g = 95°, Reason: Exterior angle of a triangle equals sum of opposite interior angles.

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## ✔ Final Answers Summary:

| Angle | Value | Reason |
|------|-------|--------|
| a | 72° | Opposite angles in a parallelogram are equal |
| b | 113° | Adjacent angles in a parallelogram are supplementary |
| c | 106° | Exterior angle = 180° – interior angle; interior angle found via quadrilateral total 360° |
| d | 143° | Sum of angles in triangle = 180° → third angle = 37°; then 180° – 37° = 143° |
| e | 81° | Corresponding angles are equal (parallel lines) |
| f | 42° | Angles in a triangle sum to 180° |
| g | 95° | Exterior angle = sum of opposite interior angles (21° + 74°) |

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