Angles and Lines

Starting with lines

Geometry begins with some simple ideas. A point is just a tiny dot that marks a position. When points join up and stretch out, we get lines.

• A line is straight and goes on forever in both directions.
• A line segment is a straight piece of a line with two end points — like the edge of a ruler.
• A ray starts at one point and goes on forever in one direction, like a beam from a torch.

Lines can be related to each other in special ways.

Parallel and perpendicular lines

Two important kinds of line relationships have special names.

Parallel lines run alongside each other and never meet, no matter how far they go. They always stay the same distance apart. Think of railway tracks or the two long edges of a ruler. We sometimes mark parallel lines with little arrows.

Perpendicular lines cross each other at a right angle (a perfect square corner). Think of the lines on a window pane, or the corner where a wall meets the floor.

Relationship	What it looks like	Real-world example
Parallel	Same distance apart, never touch	Railway tracks, ladder sides
Perpendicular	Cross at a square corner	Window frame, letter "T"

What is an angle?

When two lines, rays or segments meet at a point, they make an angle. An angle measures the amount of turn between them. The point where they meet is called the vertex, and the two lines are the arms.

We measure angles in degrees, written with a small circle: °. Here is the key reference to remember:

• A full turn all the way around is 360°.
• A half turn, making a straight line, is 180°.
• A quarter turn, making a square corner, is 90°.

Why a full turn is 360°

You might wonder why a full circle is 360 degrees and not, say, 100. The number 360 comes from the ancient Babylonians thousands of years ago. They liked the number 360 because it divides evenly by so many numbers — 2, 3, 4, 5, 6, 8, 9, 10 and more. That makes it easy to split a turn into halves, thirds, quarters and other neat fractions, which is very handy.

The types of angle

Angles are sorted by their size. Learn these five and you can name almost any angle you see.

Angle type	Size	Example
Acute	Less than 90°	30°, 45°, 80°
Right	Exactly 90°	The corner of a book
Obtuse	Between 90° and 180°	100°, 130°, 170°
Straight	Exactly 180°	A flat, straight line
Reflex	Between 180° and 360°	200°, 300°

A handy memory trick: an acute angle is small and "a c-ute little angle". An obtuse angle is big and open.

Worked examples

Example 1 — Naming angles. Sort these angles into types: 25°, 90°, 145°, 180°, 250°.

1. 25° is less than 90°, so it is acute.
2. 90° is exactly 90°, so it is a right angle.
3. 145° is between 90° and 180°, so it is obtuse.
4. 180° is a straight angle.
5. 250° is between 180° and 360°, so it is reflex.

Example 2 — Angles on a straight line. Two angles sit together on a straight line. One is 60°. What is the other?

1. Angles on a straight line add up to 180°.
2. So the missing angle is 180° − 60° = 120°.
3. Check: 60° is acute and 120° is obtuse — together they make a straight line. ✔

Example 3 — Angles around a point. Three angles meet at a single point and fill the whole turn. Two of them are 150° and 90°. Find the third.

1. Angles around a point add up to 360°.
2. So far we have 150° + 90° = 240°.
3. The missing angle is 360° − 240° = 120°.

Measuring with a protractor

To measure an angle accurately, use a protractor:

1. Place the centre hole of the protractor exactly on the vertex.
2. Line up the zero line with one arm of the angle.
3. Follow the other arm out to the scale and read the number.
4. Choose the scale (inner or outer) that starts at zero on your first arm — this avoids the common mistake of reading the wrong row.

A practice activity

Go on an "angle hunt" around your home:

1. Find three right angles (try the corner of a door, a table, or a book).
2. Find one acute angle (the hands of a clock at 1 o'clock make a small one).
3. Find one obtuse angle (an open laptop, or a clock at 8 o'clock).
4. Spot a pair of parallel lines and a pair of perpendicular lines in the room.

Then draw a square and a triangle and mark every angle you can find inside them.

Where this leads

Angles and lines are the foundation of all geometry. They lead naturally into the study of 2D and 3D Shapes, where you measure the angles inside triangles and squares, and into Area and Perimeter, where right angles help you measure flat shapes. Keep looking for angles in the world around you — buildings, sports pitches and road signs are full of them.