Angles in a Triangle

The most important rule about triangles

Here is one of the most useful facts in all of geometry: the three interior angles of any triangle always add up to exactly 180°. It does not matter whether the triangle is large or tiny, tall and thin or short and wide, right-angled, acute or obtuse — the three angles inside always total 180°. This is called the angle sum of a triangle.

(An interior angle is an angle inside the triangle, at one of its corners, or vertices.)

This single rule is incredibly powerful, because it means that if you know two of a triangle's angles, you can always work out the third one. It also helps explain the special angles in the different types of triangle, which you can review in our lesson on types of triangles.

A diagram in words: why 180°?

Here is a way to see why the rule is true. Imagine a triangle sitting on a table, with its three corners labelled A, B and C, and its three angles coloured red, green and blue.

Now picture tearing off the three corners. Take the red corner and place it on a flat line. Place the green corner right next to it, sharing an edge. Then place the blue corner next to the green one. The three torn corners fit together perfectly along a straight line, with no gaps and no overlaps. A straight line is 180° — so the three angles must add up to 180°. Every triangle you ever tear up will do the same thing. This hands-on demonstration is also the activity at the end of this lesson.

Finding a missing angle

Because the three angles total 180°, finding a missing angle is simple subtraction:

Missing angle = 180° − (sum of the two known angles)

Worked example 1: a basic missing angle

A triangle has angles of 45° and 85°. Find the third angle.

Add the two known angles: 45 + 85 = 130°. Subtract from 180: 180 − 130 = 50°. The third angle is 50°. Check: 45 + 85 + 50 = 180. ✓

Worked example 2: a right-angled triangle

A right-angled triangle has one angle of 90° and another of 32°. Find the third angle.

Add the known angles: 90 + 32 = 122°. Subtract from 180: 180 − 122 = 58°. The third angle is 58°. Notice that in any right-angled triangle the two non-right angles must add up to 90°, because the right angle already uses 90° of the 180°.

Angles in special triangles

The angle-sum rule explains the angles in the special triangles you already know about. (Need a refresher on right, acute and obtuse angles? See angles and lines.)

Equilateral triangle — all three sides are equal, so all three angles are equal. Since they share 180° equally, each angle is 180 ÷ 3 = 60°.

Isosceles triangle — two sides are equal, and the two base angles (the angles opposite the equal sides) are also equal. This lets you find all the angles even when you only know one.

Worked example 3: isosceles triangle

An isosceles triangle has an apex angle (the angle between the two equal sides) of 50°. Find the two base angles.

The apex angle uses 50° of the total 180°, leaving 180 − 50 = 130° to be shared between the two equal base angles. Divide by 2: 130 ÷ 2 = 65°. Each base angle is 65°. Check: 50 + 65 + 65 = 180. ✓

Worked example 4: the other isosceles case

An isosceles triangle has one base angle of 70°. Find the other two angles.

The base angles are equal, so the other base angle is also 70°. Together the two base angles use 70 + 70 = 140°. The apex angle is 180 − 140 = 40°. Check: 70 + 70 + 40 = 180. ✓

The exterior angle rule

If you extend one side of a triangle outward past a vertex, you create an exterior angle between the extended line and the next side. There is a beautifully simple rule:

An exterior angle equals the sum of the two interior angles it is not next to (called the two "remote" interior angles).

Worked example 5: using the exterior angle

A triangle has two interior angles of 55° and 65°. The side at the third vertex is extended to make an exterior angle. Find that exterior angle.

The exterior angle equals the sum of the two remote interior angles (the ones it is not beside): 55 + 65 = 120°. The exterior angle is 120°.

You can check this another way. The third interior angle is 180 − 55 − 65 = 60°. The exterior angle sits on a straight line with that interior angle, so it is 180 − 60 = 120°. Both methods agree.

Why this matters

The angle-sum rule is a cornerstone of geometry, and you will use it constantly. It lets you find unknown angles in any figure that can be broken into triangles — which is almost every shape, since any polygon can be split into triangles. (That is how we know a quadrilateral's angles total 360°: it splits into two triangles, 2 × 180°.) Beyond the classroom, surveyors use triangle angles to measure land and heights they cannot reach directly, a technique called triangulation. Builders check that structures are true, navigators fix their position from angles to landmarks or stars, and engineers rely on the rigidity of triangles in bridges and roofs. Mastering how angles behave inside a triangle gives you a reliable tool for solving a huge range of real problems.

Activity: prove it yourself

1. The tear test: cut out a large paper triangle of any shape — make it deliberately lopsided. Colour each of the three corners a different colour.
2. Tear off the three coloured corners.
3. Arrange the three torn corners so their points all meet at one spot and their straight edges line up. They will form a perfectly straight line with no gaps. A straight line is 180° — you have just shown the angles add to 180°!
4. Repeat with a totally different triangle (try a long thin one and a right-angled one). It works every time.
5. Measure and check: use a protractor to measure all three angles of two more triangles and add them up. Because of small measuring errors you might get 179° or 181°, but you should land right around 180°.
6. Challenge: draw a four-sided shape (quadrilateral) and split it into two triangles with one diagonal line. Use the 180° rule to explain why a quadrilateral's angles must add up to 360°.