# Parallel lines, diagonals and quadrilaterals · Year 5

> Parallel lines, diagonals and quadrilaterals, Year 5: Diagonals and more. Explanation, worked examples and free practice for the national curriculum in England.

In Parallel lines, diagonals and quadrilaterals (Year 5) you practise 4 skills: Parallel and perpendicular · Diagonals · Quadrilaterals and their properties · Line symmetry.

Two parallel lines go the same way and never meet, however far you extend them. Two perpendicular lines meet at a right angle, like the corner of a square on squared paper. Lines that are not parallel cross, even if only outside the picture.

A diagonal is a segment that joins two vertices of a polygon that are not next to each other. From each vertex, diagonals go to every vertex except itself and its two neighbours: in a pentagon, 2 diagonals start at each vertex. To count all the diagonals, multiply the number of vertices by the number from each vertex and divide by 2, because each diagonal is counted twice: in a pentagon 5 × 2 ÷ 2 = 5.

Quadrilaterals have properties of their sides, angles and diagonals. In a square and a rhombus all the sides are equal. A square, a rectangle, a rhombus and a parallelogram have two pairs of parallel sides, and a trapezium has exactly one pair. In a square and a rectangle all the angles are right angles and the diagonals are equal in length; in a square and a rhombus the diagonals are perpendicular.

A line of symmetry cuts a shape into two parts that match like a reflection in a mirror: fold along it and the parts land exactly on each other. In a square and a rhombus a diagonal is a line of symmetry.

**Tip:** To check whether two sides are parallel, count on the squares how far each one moves across and up.

## What you learn

### Parallel and perpendicular

- Example: (Two straight lines on squared paper.) Are the lines parallel, perpendicular, or meeting but not perpendicular?
- Options: parallel · meeting, but not perpendicular · perpendicular
- Answer: parallel
- Solution: The lines go the same way: at every step both move the same number of squares across and up, so they never meet. They are parallel.

### Diagonals

- Example: (A quadrilateral with its vertices marked by letters.) Which segment is a diagonal of the polygon?
- Options: BD · AD · CD
- Answer: BD
- Solution: (A quadrilateral with its vertices marked by letters.) BD . It joins two vertices that are not next to each other. The other segments are sides of the polygon.

### Quadrilaterals and their properties

- Example: (A square.) Is it true that in the square in the picture the diagonals are perpendicular?
- Options: Yes · No
- Answer: Yes
- Solution: Yes: in the square in the picture the diagonals are perpendicular.

### Line symmetry

- Example: (An equilateral triangle with a dashed line through it) Is the dashed line a line of symmetry of the shape?
- Options: Yes · No
- Answer: Yes
- Solution: When you fold along the dashed line, the two parts land exactly on each other. So yes, it is a line of symmetry.

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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