Angles in Parallel Lines

What you'll learn

How to use key angle facts: straight lines, vertically opposite angles and triangles.
How to spot corresponding, alternate and co-interior angles.
How to combine parallel-line rules with isosceles triangles and parallelograms.
How to write clear reasons for your working.

1. Angle facts you already need

Before parallel lines, make sure these basic angle facts are secure:

Angles on a straight line add to 180°.
Vertically opposite angles are the pair of angles opposite each other when two straight lines cross. They are equal.
Angles in a triangle add to 180°.

Example

Using a straight line and vertically opposite angles

Two straight lines cross. One angle is 68°. Find the angle opposite it and one angle next to it.

Two intersecting straight lines showing the 68° angle, its vertically opposite equal angle, and an adjacent angle on a straight line.

The angle opposite 68° is also 68°, because vertically opposite angles are equal.
The angle next to 68° lies on a straight line with it, so they add to 180°.
The next angle is $180∘−68∘=112∘180^\circ - 68^\circ = 112^\circ$ .

2. Parallel lines and transversals

Definition

Parallel lines and a transversal

Parallel lines are straight lines that stay the same distance apart and never meet. A transversal is a straight line that cuts across two or more other lines. We write $\parallel CD$ to mean line $A B$ is parallel to line $C D$ .

Parallel lines AB and CD cut by a transversal, with matching arrows marking the parallel lines and the interior region between them.

On diagrams, parallel lines are often marked with matching arrows. The interior means the space between the two parallel lines.

3. Corresponding angles: the F-shape

Corresponding angles are angles in the same relative position at each crossing. For example, both might be in the “top-right corner” where the transversal crosses the two parallel lines.

Key Idea

Corresponding angles

When a transversal crosses parallel lines, corresponding angles are equal. Look for an F-shape.

Example

Writing down angles with reasons

Two parallel lines are crossed by a transversal. At the top crossing, the bottom-right angle is 58°. Angle $x$ is top-left at the same crossing. Angle $y$ is top-left at the lower crossing.

Corresponding angles in the same top-left position at each crossing, with a vertically opposite 58° angle used to find x first.

Angle $x$ is opposite 58° at the same crossing, so $58^\circ$ because vertically opposite angles are equal.
Angle $y$ is in the same top-left position as angle $x$ on the other parallel line.
Therefore $58^\circ$ because corresponding angles are equal.

4. Alternate angles: the Z-shape

Alternate angles are on opposite sides of the transversal. They often sit inside the parallel lines and make a Z-shape.

Key Idea

Alternate angles

When a transversal crosses parallel lines, alternate angles are equal. Look for a Z-shape.

Example

Finding an alternate angle

Line $A B$ is parallel to line $C D$ . A transversal cuts them. An interior angle below $A B$ on the left of the transversal is 34°. Angle $x$ is above $C D$ on the right of the transversal.

Alternate interior angles on opposite sides of the transversal forming a Z-shape.

The two angles are inside the parallel lines and on opposite sides of the transversal.
This is a Z-shape, so the angles are alternate angles.
Therefore $34^\circ$ because alternate angles are equal.

5. Co-interior angles: the C-shape

Co-interior angles are inside the parallel lines and on the same side of the transversal.

Key Idea

Co-interior angles

Co-interior angles add to 180°. Look for a C-shape.

Example

Using angles that add to 180°

Two parallel lines are cut by a transversal. One co-interior angle is 118°. The other co-interior angle is $x$ .

Co-interior angles inside the parallel lines on the same side of the transversal, forming a C-shape and adding to 180°.

The two angles are inside the parallel lines and on the same side of the transversal.
This is a C-shape, so they are co-interior angles.
Co-interior angles add to 180°, so $180^\circ - 118^\circ = 62^\circ$ .

Common Mistake

Equal or add to 180?

Corresponding and alternate angles are equal, but co-interior angles add to 180°. If you see a C-shape, do not copy the angle across unchanged.

6. Combining with triangles

In harder questions, you often need more than one fact. Work one angle at a time and give a reason each time.

Definition

Isosceles triangle

An isosceles triangle has two equal sides. The angles opposite those equal sides are equal.

Example

Parallel lines plus an isosceles triangle

Line $A B$ is parallel to line $C D$ . Points E and F lie on $A B$ , and point G lies on $C D$ . The triangle EFG is isosceles with $E G = F G$ . The outside angle AEG is 112°. Find $∠FGD\angle FGD$ .

An isosceles triangle between parallel lines, with the outside angle at E and the target angle FGD at G.

Angle AEG and angle GEF lie on a straight line, so $∠GEF=180∘−112∘=68∘\angle GEF = 180^\circ - 112^\circ = 68^\circ$ .
Since $E G = F G$ , triangle EFG is isosceles. The base angles are equal, so $∠EFG=68∘\angle EFG = 68^\circ$ .
Line $E F$ is parallel to line $G D$ , and $F G$ is a transversal.
Therefore $∠FGD=68∘\angle FGD = 68^\circ$ because alternate angles are equal.

Tip

Angle chasing

Angle chasing means finding unknown angles one by one. Write the new angle size on the diagram, then write the reason next to your working.

7. Parallelograms: hidden parallel lines

Definition

Parallelogram

A parallelogram is a four-sided shape with both pairs of opposite sides parallel. In parallelogram $A B C D$ , this means $\parallel CD$ and $\parallel BC$ .

Parallelogram ABCD with both pairs of opposite sides marked parallel.

A parallelogram question is usually a parallel-lines question in disguise. Sometimes a side is extended, or a diagonal is drawn. A diagonal is a line joining opposite corners.

Example

Using a parallelogram and a triangle

ABCD is a parallelogram. Point E lies on the same straight line as B and C. Angle DAB is 126° and angle AEB is 38°. Find $∠BAE\angle BAE$ .

A parallelogram with BC extended to E, forming triangle ABE and showing the given angles used to find angle BAE.

In a parallelogram, opposite sides are parallel, so $\parallel BC$ .
Since E lies on the same straight line as B and C, line $B E$ is also parallel to line $A D$ .
Angles DAB and ABE are co-interior angles, so they add to 180°. Therefore $∠ABE=180∘−126∘=54∘\angle ABE = 180^\circ - 126^\circ = 54^\circ$ .
In triangle ABE, angles add to 180°, so $∠BAE=180∘−54∘−38∘=88∘\angle BAE = 180^\circ - 54^\circ - 38^\circ = 88^\circ$ .

Exam technique

In the exam

Mark every known angle on the diagram before calculating.
Write a reason next to each new angle: straight line, vertically opposite, corresponding, alternate, co-interior, triangle total, isosceles or parallelogram.
If two lines are marked as parallel, look for F, Z or C shapes before doing harder triangle work.

Self review

Check yourself

Which angle rule gives equal angles in an F-shape?
Which angle rule gives angles that add to 180° in a C-shape?
In an isosceles triangle, where are the equal angles compared with the equal sides?

Angles in Parallel Lines

What you'll learn

How to use key angle facts: straight lines, vertically opposite angles and triangles.
How to spot corresponding, alternate and co-interior angles.
How to combine parallel-line rules with isosceles triangles and parallelograms.
How to write clear reasons for your working.

1. Angle facts you already need

Before parallel lines, make sure these basic angle facts are secure:

Angles on a straight line add to 180°.
Vertically opposite angles are the pair of angles opposite each other when two straight lines cross. They are equal.
Angles in a triangle add to 180°.

Example

Using a straight line and vertically opposite angles

Two straight lines cross. One angle is 68°. Find the angle opposite it and one angle next to it.

Two intersecting straight lines showing the 68° angle, its vertically opposite equal angle, and an adjacent angle on a straight line.

The angle opposite 68° is also 68°, because vertically opposite angles are equal.
The angle next to 68° lies on a straight line with it, so they add to 180°.
The next angle is $180∘−68∘=112∘180^\circ - 68^\circ = 112^\circ$ .

2. Parallel lines and transversals

Definition

Parallel lines and a transversal

Parallel lines AB and CD cut by a transversal, with matching arrows marking the parallel lines and the interior region between them.

On diagrams, parallel lines are often marked with matching arrows. The interior means the space between the two parallel lines.

3. Corresponding angles: the F-shape

Corresponding angles are angles in the same relative position at each crossing. For example, both might be in the “top-right corner” where the transversal crosses the two parallel lines.

Key Idea

Corresponding angles

When a transversal crosses parallel lines, corresponding angles are equal. Look for an F-shape.

Example

Writing down angles with reasons

Two parallel lines are crossed by a transversal. At the top crossing, the bottom-right angle is 58°. Angle $x$ is top-left at the same crossing. Angle $y$ is top-left at the lower crossing.

Corresponding angles in the same top-left position at each crossing, with a vertically opposite 58° angle used to find x first.

Angle $x$ is opposite 58° at the same crossing, so $58^\circ$ because vertically opposite angles are equal.
Angle $y$ is in the same top-left position as angle $x$ on the other parallel line.
Therefore $58^\circ$ because corresponding angles are equal.

4. Alternate angles: the Z-shape

Alternate angles are on opposite sides of the transversal. They often sit inside the parallel lines and make a Z-shape.

Key Idea

Alternate angles

When a transversal crosses parallel lines, alternate angles are equal. Look for a Z-shape.

Example

Finding an alternate angle

Line $A B$ is parallel to line $C D$ . A transversal cuts them. An interior angle below $A B$ on the left of the transversal is 34°. Angle $x$ is above $C D$ on the right of the transversal.

Alternate interior angles on opposite sides of the transversal forming a Z-shape.

The two angles are inside the parallel lines and on opposite sides of the transversal.
This is a Z-shape, so the angles are alternate angles.
Therefore $34^\circ$ because alternate angles are equal.

5. Co-interior angles: the C-shape

Co-interior angles are inside the parallel lines and on the same side of the transversal.

Key Idea

Co-interior angles

Co-interior angles add to 180°. Look for a C-shape.

Example

Using angles that add to 180°

Two parallel lines are cut by a transversal. One co-interior angle is 118°. The other co-interior angle is $x$ .

Co-interior angles inside the parallel lines on the same side of the transversal, forming a C-shape and adding to 180°.

The two angles are inside the parallel lines and on the same side of the transversal.
This is a C-shape, so they are co-interior angles.
Co-interior angles add to 180°, so $180^\circ - 118^\circ = 62^\circ$ .

Common Mistake

Equal or add to 180?

Corresponding and alternate angles are equal, but co-interior angles add to 180°. If you see a C-shape, do not copy the angle across unchanged.

6. Combining with triangles

In harder questions, you often need more than one fact. Work one angle at a time and give a reason each time.

Definition

Isosceles triangle

An isosceles triangle has two equal sides. The angles opposite those equal sides are equal.

Example

Parallel lines plus an isosceles triangle

An isosceles triangle between parallel lines, with the outside angle at E and the target angle FGD at G.

Angle AEG and angle GEF lie on a straight line, so $∠GEF=180∘−112∘=68∘\angle GEF = 180^\circ - 112^\circ = 68^\circ$ .
Since $E G = F G$ , triangle EFG is isosceles. The base angles are equal, so $∠EFG=68∘\angle EFG = 68^\circ$ .
Line $E F$ is parallel to line $G D$ , and $F G$ is a transversal.
Therefore $∠FGD=68∘\angle FGD = 68^\circ$ because alternate angles are equal.

Tip

Angle chasing

Angle chasing means finding unknown angles one by one. Write the new angle size on the diagram, then write the reason next to your working.

7. Parallelograms: hidden parallel lines

Definition

Parallelogram

A parallelogram is a four-sided shape with both pairs of opposite sides parallel. In parallelogram $A B C D$ , this means $\parallel CD$ and $\parallel BC$ .

Parallelogram ABCD with both pairs of opposite sides marked parallel.

A parallelogram question is usually a parallel-lines question in disguise. Sometimes a side is extended, or a diagonal is drawn. A diagonal is a line joining opposite corners.

Example

Using a parallelogram and a triangle

ABCD is a parallelogram. Point E lies on the same straight line as B and C. Angle DAB is 126° and angle AEB is 38°. Find $∠BAE\angle BAE$ .

A parallelogram with BC extended to E, forming triangle ABE and showing the given angles used to find angle BAE.

In a parallelogram, opposite sides are parallel, so $\parallel BC$ .
Since E lies on the same straight line as B and C, line $B E$ is also parallel to line $A D$ .
Angles DAB and ABE are co-interior angles, so they add to 180°. Therefore $∠ABE=180∘−126∘=54∘\angle ABE = 180^\circ - 126^\circ = 54^\circ$ .
In triangle ABE, angles add to 180°, so $∠BAE=180∘−54∘−38∘=88∘\angle BAE = 180^\circ - 54^\circ - 38^\circ = 88^\circ$ .

Exam technique

In the exam

Mark every known angle on the diagram before calculating.
Write a reason next to each new angle: straight line, vertically opposite, corresponding, alternate, co-interior, triangle total, isosceles or parallelogram.
If two lines are marked as parallel, look for F, Z or C shapes before doing harder triangle work.

Self review

Check yourself

Which angle rule gives equal angles in an F-shape?
Which angle rule gives angles that add to 180° in a C-shape?
In an isosceles triangle, where are the equal angles compared with the equal sides?

Angles in Parallel Lines

What you'll learn

1. Angle facts you already need

2. Parallel lines and transversals

3. Corresponding angles: the F-shape

4. Alternate angles: the Z-shape

5. Co-interior angles: the C-shape

6. Combining with triangles

7. Parallelograms: hidden parallel lines

Recap questions

You've reached the end

Angles in Parallel Lines

What you'll learn

1. Angle facts you already need

2. Parallel lines and transversals

3. Corresponding angles: the F-shape

4. Alternate angles: the Z-shape

5. Co-interior angles: the C-shape

6. Combining with triangles

7. Parallelograms: hidden parallel lines

Recap questions

You've reached the end

Angles in Parallel Lines

What you'll learn

1. Angle facts you already need

2. Parallel lines and transversals

3. Corresponding angles: the F-shape

4. Alternate angles: the Z-shape

5. Co-interior angles: the C-shape

6. Combining with triangles

7. Parallelograms: hidden parallel lines

Recap questions

You've reached the end

Angles in Parallel Lines Revision Guide

Angles in Parallel Lines

What you'll learn

1. Angle facts you already need

2. Parallel lines and transversals

3. Corresponding angles: the F-shape

4. Alternate angles: the Z-shape

5. Co-interior angles: the C-shape

6. Combining with triangles

7. Parallelograms: hidden parallel lines

Recap questions

You've reached the end

Angles in Parallel Lines Revision Guide