Why Do the Angles in a Triangle Add Up to 180°?

A proof you can do with paper, a proof with parallel lines, and how the 180° rule, together with isosceles and equilateral triangles, solves triangle angle problems.

3 min read Supports 3 lessons

Cut a triangle out of paper. Tear off its three corners and line them up, point to point. However long, thin or lopsided the triangle, the corners always fit together into a straight line: $180°$ .

$50° + 60° + 70° = 180°$ .

Triangle angle sum

a + b + c = 180^\circ

Why it is always true

Draw a line through the top corner, parallel to the base. The angles along that line form a straight line, so they add to $180°$ . The two outer ones are alternate angles, equal to the triangle's base angles. So the three angles of the triangle are exactly the three angles on the straight line.

$\text{Draw a line through the apex, parallel to the base.}$

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Special triangles

Key idea

Isosceles: two equal sides and two equal base angles. Equilateral: all sides equal, all angles $60°$ .

Worked example

An isosceles triangle has an apex angle of $40°$ . Find each base angle.

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The two base angles share what is left.
$180^\circ - 40^\circ = 140^\circ$

Isosceles: equal base angles.

Right-angled triangles

If one angle is $90°$ , the other two must add to $90°$ . That simple fact is behind much of trigonometry, where knowing one acute angle tells you the other.

The exterior angle

Tip

An exterior angle of a triangle (made by extending one side) equals the sum of the two interior angles opposite it. Both it and those two angles make $180°$ with the same third angle.

Polygons

Any polygon can be split into triangles from one corner. A shape with $n$ sides splits into $n - 2$ triangles, so its interior angles add to $(n - 2) \times 180°$ . A quadrilateral's add to $360°$ ; a pentagon's to $540°$ .

Watch out

The $180°$ rule is for flat surfaces. On a sphere, such as the Earth, a triangle's angles add to more than $180°$ .

Common questions

Do the angles in every triangle add to 180°?: Yes, for any triangle drawn on a flat surface, whatever its shape or size.
How do I find the angles in an isosceles triangle?: The two base angles are equal. Subtract the known angle from $180°$ , then share the rest equally if the unknowns are the base angles.

Practise triangle angle sum

The Triangle Angles lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Do the angles in every triangle add to 180°?

Yes, for any triangle drawn on a flat surface, whatever its shape or size.

How do I find the angles in an isosceles triangle?

The two base angles are equal. Subtract the known angle from $180°$ , then share the rest equally if the unknowns are the base angles.