Angles in Parallel Lines Textbook Exercise - Corbettmaths - Free Printable

Let's solve each part of Question 1 step by step, using properties of parallel lines and angles:

We’ll use the following key angle rules:
- Corresponding angles are equal (same position on parallel lines).
- Alternate angles are equal (Z-shape).
- Co-interior (or consecutive) angles add up to 180° (C-shape).
- Vertically opposite angles are equal.
- Angles on a straight line sum to 180°.
- Around a point: 360°.

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(a)

Given:
Two parallel lines cut by a transversal.
One angle is 112°, and we need to find x.

The 112° angle and x are vertically opposite at the same intersection — so they are equal.

✔ x = 112°

> But wait! Let’s double-check: Is x vertically opposite?

Yes — both angles are on the same side of the transversal, and directly across from each other. So yes, vertically opposite angles are equal.

✔ x = 112°

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(b)

Given:
Top angle = 75°, find x.

This is a transversal cutting two parallel lines.

The 75° angle and x are corresponding angles (they're in the same relative position).

So, corresponding angles are equal.

✔ x = 75°

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(c)

Given:
One angle is 150°, and we need to find x and y.

Look at the diagram:
- The 150° angle and x are on a straight line → they form a linear pair.

So:
x + 150° = 180°
→ x = 30°

Now, x = 30° and y are corresponding angles (same position on the parallel lines), so:

✔ y = 30°

> Alternatively, you could say y is alternate to the 150° angle? Wait — no. Let’s check:

Actually, the 150° angle and y are on the same side of the transversal and between the lines — that makes them co-interior angles. But co-interior angles add to 180° only if the lines are parallel — which they are.

Wait — let’s re-analyze:

- The 150° angle is on the bottom line, inside, on the right.
- y is on the top line, inside, on the right → so they are corresponding angles.

But 150° ≠ y → unless y = 150°? But earlier we said x = 30°.

Wait — there's a contradiction.

Let’s fix this.

Look carefully:

- The angle marked 150° is on the bottom line, on the left side of the transversal, inside the parallel lines.
- x is on the bottom line, but on the right side of the transversal — so it's adjacent to 150°.

So, x + 150° = 180° → x = 30°

Now, x = 30° is on the bottom line, right side.

Now, y is on the top line, right side — same side as x → so y and x are corresponding angles → so y = x = 30°

✔ So:
- x = 30°
- y = 30°

✔️ Correct.

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(d)

Given:
Two vertical lines (parallel), a transversal crosses them, forming angles.

One angle is 99°, and we need x, y, z.

Let’s analyze:

- The 99° angle and x are adjacent on a straight line → so:
- x + 99° = 180° → x = 81°

Now, x = 81° and y are vertically opposite angles → so y = 81°

Also, z is vertically opposite to the 99° angle → so z = 99°

Alternatively, z and 99° are vertically opposite → yes.

✔ So:
- x = 81°
- y = 81°
- z = 99°

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(e)

Three parallel lines, one transversal.

Top angle: 74°

Find x and y.

- The 74° angle and x are corresponding angles → so x = 74°

- Now, x and y are corresponding angles again (same position on next parallel line) → so y = 74°

✔ So:
- x = 74°
- y = 74°

> Note: Since all lines are parallel and the transversal cuts them equally, all corresponding angles are equal.

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(f)

Complex diagram with multiple lines.

We have:
- Top left: 123°
- Bottom right: 110°
- Need to find x, y

Let’s break it down.

First, look at the 123° angle.

It’s on the top line, and it's interior, on the left side.

Its vertically opposite angle is also 123°.

Now, consider the angle adjacent to 123° on the same line → that would be:
- 180° – 123° = 57°

Now, this 57° angle is corresponding to x (since both are on the same side of the transversal and in the same relative position).

So, x = 57°

Now, look at the 110° angle on the bottom line.

It’s on the right side, interior.

Its adjacent angle on the same line is:
- 180° – 110° = 70°

Now, this 70° angle is corresponding to y?

Wait — let’s see.

The 110° angle is on the bottom line, right side, interior.

The y angle is on the top line, right side, interior.

Are they corresponding? Yes — same position.

So, y = 110°

But wait — is that correct?

Wait — the transversal goes from top-left to bottom-right.

So, the 110° angle and y are on the same side of the transversal and between the lines → so they are co-interior angles.

But co-interior angles add to 180° when lines are parallel.

So if the lines are parallel, then:

y + 110° = 180° → y = 70°

Ah! That’s the key.

So, y = 70°

But earlier I thought y was corresponding — but actually, since both are on the same side of the transversal and between the lines, they are co-interior, not corresponding.

So:
- y + 110° = 180° → y = 70°

Now, what about x?

We had 123° at the top left.

The angle adjacent to it is 180° – 123° = 57°

Now, this 57° angle is corresponding to x?

Let’s see:

- The 57° angle is on the top line, right side of the transversal.
- x is on the bottom line, left side of the transversal.

No — not corresponding.

Wait — better way:

Look at the 123° angle. It’s on the top line, left side, interior.

Its vertically opposite angle is also 123°.

Now, x is on the bottom line, left side, interior.

So, x and the 123° angle are corresponding? Yes — same position.

So, x = 123°

Wait — but earlier I said x = 57° — that was wrong.

Let’s clarify.

If x is on the bottom line, left side, interior, and the 123° is on the top line, left side, interior, then they are corresponding angles → so x = 123°

But wait — the 123° is on the top line, and x is on the bottom line — so yes, same side, same position → corresponding → x = 123°

But now, is that consistent?

Let’s check the other transversal.

There is another transversal on the right.

We have 110° on the bottom line, right side, interior.

And y is on the top line, right side, interior.

So, y and 110° are co-interior angles → so:

y + 110° = 180° → y = 70°

Now, is there any conflict?

But wait — can both x = 123° and y = 70°?

Yes — different transversals.

So, x comes from the left transversal, y from the right transversal.

So:

✔ x = 123° (corresponding to 123°)
✔ y = 70° (co-interior with 110°)

But wait — let’s confirm x.

Is x really corresponding to 123°?

Yes:
- Both are interior angles,
- On the same side of the transversal (left),
- One on top line, one on bottom line → so yes, corresponding → x = 123°

And for y:
- 110° and y are both interior, on the right side of the right transversal,
- So they are co-interior → sum to 180° → y = 70°

✔ So final answers:

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✔ Final Answers:

(a) x = 112°
(b) x = 75°
(c) x = 30°, y = 30°
(d) x = 81°, y = 81°, z = 99°
(e) x = 74°, y = 74°
(f) x = 123°, y = 70°

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Summary Table:

| Part | x | y | z |
|------|---------|---------|-------|
| (a) | 112° | | |
| (b) | 75° | | |
| (c) | 30° | 30° | |
| (d) | 81° | 81° | 99° |
| (e) | 74° | 74° | |
| (f) | 123° | 70° | |

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