# Parallel lines and trapezia · Years 8–9

> Parallel lines and trapezia, Years 8–9: Angles in a trapezium and more. Explanation, worked examples and free practice for the national curriculum in England.

In Parallel lines and trapezia (Years 8–9) you practise 4 skills: When are two lines parallel? · Angles in a trapezium · Area and perimeter of a trapezium · Is it a trapezium? On a coordinate grid.

Two lines crossed by a third line are parallel if a pair of corresponding angles is equal, or a pair of alternate angles is equal, or the two angles between the lines on the same side of the third line add up to 180°. This is the converse of the angle facts for parallel lines: there the lines were known to be parallel and you found the angles; here the angles tell you that the lines are parallel.

A trapezium is a quadrilateral with exactly one pair of parallel sides. The other two sides are its legs. The two angles at the same leg add up to 180°, because they lie between the parallel sides on the same side of the leg. In an isosceles trapezium the legs are equal and the base angles are equal.

The area of a trapezium is the sum of the parallel sides times the height, divided by 2. The height is the distance between the parallel sides, at right angles to both. On a coordinate grid, two sides are parallel when their gradients are equal, or when both are vertical.

**Tip:** To check whether a quadrilateral is a trapezium, test both pairs of opposite sides: in a trapezium exactly one pair is parallel.

## What you learn

### When are two lines parallel?

- Example: The picture shows two lines and a third line crossing them. (Two lines, and a third line crossing both. The angles it makes are numbered 1 to 8.) Given ∢1 = 75°, ∢5 = 75° . Are the lines parallel?
- Options: Yes · No
- Answer: Yes
- Solution: Angles 1 and 5 are corresponding angles, and they are equal. So the lines are parallel.

### Angles in a trapezium

- Example: (A trapezium: the long base at the bottom, the short base above it and parallel to it.) In the trapezium ABCD the bases are parallel: AB ∥ DC . Given ∢A = 45° . Find ∢D .
- Answer: 135 °
- Solution: At the leg AD : ∢D = 180° − 45° = 135°

### Area and perimeter of a trapezium

- Example: (A trapezium: the long base at the bottom, the short base above it and parallel to it. The height to the base is dashed.) The parallel sides of the trapezium are 7 cm and 5 cm, and its height is 4 cm. What is its area?
- Answer: 24 cm²
- Solution: (7 + 5) × 4 ÷ 2 = 24

### Is it a trapezium? On a coordinate grid

- Example: (A coordinate grid with the quadrilateral ABCD) The points are A(1, −6), B(5, −6), C(7, −1), D(−1, 0) . Is the quadrilateral ABCD a trapezium?
- Options: Yes · No
- Answer: No
- Solution: The gradient of AB is 0 ÷ 4 = 0 The gradient of DC is −1 ÷ 8 The gradient of AD is −6 ÷ 2 = −3 The gradient of BC is 5 ÷ 2 No pair of opposite sides is parallel, so the quadrilateral is not a trapezium.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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