Right-angled triangle

01 What is always true

∠DAB means the angle at A — the middle letter is always the corner, and the outer two say which sides form it.

• No two sides have to be equal.

  AB = 70, BC = 86, CA = 50

• One of the angles is a right angle, 90°.

  ∠CAB = 90°, marked with a small square.

  Why can there only ever be one right angle?

  1. Suppose a triangle had two. Each is 90°, so between them they use 90° + 90° = 180°.

  2. But a triangle's three angles total 180° altogether. Two right angles have already spent all of it, leaving 0° for the third corner.

  3. A corner of 0° is not a corner — the two sides would lie on top of each other and the shape would never close. So two right angles are impossible, and one is the most any triangle can have.

  4. Which gives the useful version: in a right-angled triangle the other two angles always add to 180° − 90° = 90°.

• All 3 angles add up to 180°.

  True of every 3-sided shape.

02 Why it works

A triangle can have only one right angle. Two would already use up 180° between them, leaving nothing for the third corner, and a shape cannot close with a 0° angle.

So the other two angles always add to 90°. Give one of them and the other follows immediately — which is why so many questions hand you a single acute angle in a right-angled triangle.

The two short sides meeting at the right angle are the base and the height, which is what makes the area easy: half of base × height.