Angles in triangles

curriculum-key-fact
The three angles in any triangle add up to 180°.
Diagram of a scalene triangle

You can use this information to find a missing angle.

Scalene triangle

A scalene triangle has three different angles. To find a missing angle you need to know the other two angles.

Diagram of a scalene triangle with angles of 70°, 50° and an unknown angle marked as x°

\[x^\circ = 180^\circ - (70^\circ + 50^\circ )\]

\[= 180^\circ - 120^\circ = 60^\circ\]

Isosceles triangle

An isosceles triangle is one with two sides equal in length and two equal angles.

You only need to know one angle in an isosceles triangle to work out the other two.

Diagram of an isosceles triangle with the two equal angles highlighted

Question

In the diagram below the triangle is isosceles. What is the value of \(a^\circ\)?

Isosceles triangle with a 70° base angle and a vertex angle marked as a°

\[a^\circ = 180^\circ - (70^\circ + 70^\circ )\]

\[= 180^\circ - 140^\circ = 40^\circ\]

Question

What is the size of angle \({p}\)?

Isosceles triangle sum diagram

This is an isosceles triangle, so both the bottom angles are \({p}\).

The angles in a triangle add up to \({180}^\circ\), so:

\[p + p + 40 = 180\]

\[2p + 40 = 180\]

\[2p = 140\]

\[p = 70\]

So the missing angles are both \(70^\circ\).

Equilateral triangle

An equilateral triangle is one in which all three angles are equal.

The angles add up to 180°, so each angle is 60°.

You don't need to be told any angles in an equilateral to find a missing angle. The angles are always 60°.

Equilateral triangle