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 add up to 180°.
- Angles around a point add up to 360°.

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(a)

Given:
- One angle is 112°, and it’s between two parallel lines cut by a transversal.
- Angle $ x $ is below the lower line.

The 112° angle and $ x $ are corresponding angles, so they are equal.

✔ $ \boxed{x = 112^\circ} $

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(b)

Given:
- Top angle = 75°
- $ x $ is on the bottom line, same side as the 75° angle.

75° and $ x $ are corresponding angles → so they are equal.

✔ $ \boxed{x = 75^\circ} $

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(c)

Given:
- Bottom angle = 150°
- $ x $ is adjacent to it on the same line.
- $ y $ is above the top line.

First, find $ x $:
Since 150° and $ x $ are on a straight line:
$ x + 150^\circ = 180^\circ $
→ $ x = 30^\circ $

Now, $ y $ and $ x $ are corresponding angles (same relative position), so:

✔ $ \boxed{x = 30^\circ},\quad \boxed{y = 30^\circ} $

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(d)

Given:
- A transversal crosses two vertical parallel lines.
- One angle is 99°.
- We need to find $ x $, $ y $, and $ z $.

Look at the 99° angle. It's formed with the transversal and one vertical line.

Angle $ x $ is vertically opposite to 99° → so:

$ x = 99^\circ $

Now, $ x $ and $ y $ are corresponding angles (same side, same position across parallel lines), so:

$ y = 99^\circ $

Now, $ z $ is adjacent to $ y $ on a straight line → so:

$ z + y = 180^\circ $ → $ z = 180^\circ - 99^\circ = 81^\circ $

✔ $ \boxed{x = 99^\circ},\quad \boxed{y = 99^\circ},\quad \boxed{z = 81^\circ} $

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(e)

Three parallel horizontal lines, one transversal.

Top angle = 74°
We need $ x $ and $ y $

- $ x $ is corresponding to the 74° angle → so $ x = 74^\circ $
- $ y $ is alternate to the 74° angle (Z-shape) → so $ y = 74^\circ $

Wait — actually, let's check carefully.

The 74° angle is on the top line, and $ x $ is on the middle line, same side → corresponding → $ x = 74^\circ $

Then $ y $ is on the bottom line, and it’s corresponding to $ x $ → so $ y = 74^\circ $

Alternatively, you can see that $ y $ is corresponding to the original 74° angle, so also 74°.

✔ $ \boxed{x = 74^\circ},\quad \boxed{y = 74^\circ} $

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(f)

This one has two transversals and multiple angles.

Given:
- Top left angle = 123°
- Bottom right angle = 110°
- Need $ x $, $ y $

Let’s label carefully.

First, look at the 123° angle. It's on the top line, and it's co-interior with the angle just below it (on the middle line). But we don’t have that directly.

But notice: the 123° angle and $ x $ are corresponding angles if we consider the slanted transversal.

Wait — actually, let’s trace the transversals.

There are two transversals crossing the parallel lines.

Let’s break it down.

#### Step 1: Use the 123° angle

- The 123° angle is on the top line, and $ x $ is on the bottom line, but on the same transversal.
- So, $ x $ and 123° are corresponding angles → $ x = 123^\circ $? Wait — no, not necessarily.

Wait: are the lines parallel? Yes — horizontal lines are parallel.

So, the transversal cutting them: the angle at the top is 123°, and $ x $ is the angle at the bottom on the same side → corresponding, so:

✔ $ x = 123^\circ $

But wait — now check the other transversal.

On the right side, there’s an angle of 110° on the bottom line, and $ y $ is on the top line, same transversal.

So, $ y $ and 110° are corresponding angles → so $ y = 110^\circ $

But let’s double-check this.

Wait — are $ y $ and 110° corresponding?

Yes: both are on the right side, same relative position (both on the "outside" of the transversal), so yes — corresponding angles.

So:

✔ $ x = 123^\circ,\quad y = 110^\circ $

But wait — let’s verify with geometry.

Is there any contradiction?

Check: On the top line, we have a 123° angle and a $ y $ angle.

Are they on the same side? Yes.

But the angle between the two transversals on the top line must be consistent.

But since we’re told the lines are parallel and we’re using standard angle rules, and each lettered angle is defined on a different transversal, we can treat them independently.

So:

- $ x $ is corresponding to 123° → $ x = 123^\circ $
- $ y $ is corresponding to 110° → $ y = 110^\circ $

✔ $ \boxed{x = 123^\circ},\quad \boxed{y = 110^\circ} $

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✔ Final Answers:

(a) $ x = 112^\circ $
(b) $ x = 75^\circ $
(c) $ x = 30^\circ,\ y = 30^\circ $
(d) $ x = 99^\circ,\ y = 99^\circ,\ z = 81^\circ $
(e) $ x = 74^\circ,\ y = 74^\circ $
(f) $ x = 123^\circ,\ y = 110^\circ $

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