# Parallelograms · Years 8–10

> Parallelograms, Years 8–10: Angles in a parallelogram and more. Explanation, worked examples and free practice for the national curriculum in England.

In Parallelograms (Years 8–10) you practise 4 skills: Angles in a parallelogram · Sides and diagonals of a parallelogram · Is it a parallelogram? · Parallelograms on a coordinate grid.

A parallelogram is a quadrilateral whose two pairs of opposite sides are parallel. Its opposite sides are equal, its opposite angles are equal, and two adjacent angles add up to 180°.

The diagonals of a parallelogram bisect each other: where they meet is the midpoint of each. The area of a parallelogram is the base times the height to it.

A quadrilateral is a parallelogram if any one of these holds: both pairs of opposite sides are equal; one pair of opposite sides is both equal and parallel; the diagonals bisect each other; both pairs of opposite angles are equal. One pair of equal sides and the other pair parallel is not enough: an isosceles trapezium looks like that too.

**Tip:** On a coordinate grid, the step from one vertex to the next is the same as the step along the opposite side. That is how you find the fourth vertex.

## What you learn

### Angles in a parallelogram

- Example: (A parallelogram.) In the parallelogram ABCD it is given that ∢D = 145° . Find ∢A .
- Answer: 35 °
- Solution: The angles are adjacent, so they add up to 180°: ∢A = 180° − 145° = 35°

### Sides and diagonals of a parallelogram

- Example: (A parallelogram.) In the parallelogram ABCD two adjacent sides are 17 cm and 3 cm. What is its perimeter?
- Answer: 40 cm
- Solution: Each side appears twice: 2 × (17 + 3) = 40

### Is it a parallelogram?

- Example: In the quadrilateral ABCD it is given that AB ∥ DC, AD ∥ BC . Is the quadrilateral a parallelogram?
- Options: Yes, it is a parallelogram · No, it is not a parallelogram · You cannot tell
- Answer: Yes, it is a parallelogram
- Solution: A quadrilateral whose two pairs of opposite sides are parallel is a parallelogram: that is the definition.

### Parallelograms on a coordinate grid

- Example: (A coordinate grid with three vertices of a parallelogram) The points A(−5, −6), B(−3, −5), C(−5, 0) are three vertices of the parallelogram ABCD . What are the coordinates of D ?
- Answer: (−7, −1)
- Solution: From B to A you go 2 left and 1 down. The same step from C : D(−7, −1)

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).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 8 topics](https://kixplo.com/uk/en/maths/year-8/)