The Angles of a Triangle
The three angles always total 180°, whatever the size or shape. How to find a missing angle, what makes a triangle right, acute or obtuse, and why two right angles are impossible.
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One fact unlocks almost every triangle problem: the three angles always add up to 180°. It does not matter how big the triangle is or what shape it takes — the total never changes. Give me two angles and I can always find the third.
Concept
The rule
The angles of any triangle sum to 180°.
This holds for every triangle — large or small, wide or narrow. It is the single most useful fact in triangle geometry.
Example
Finding the third angle
A triangle has angles of 45° and 45°. What is the third?
⟹ 90°, so this is a right triangle
Now try angles of 30° and 60°:
⟹ 90° again — another right triangle
Concept
Three kinds of triangle
Right — one angle is exactly 90°, so the other two must add to 90°.
Acute — every angle is less than 90°.
Obtuse — one angle exceeds 90°, so the other two are both less than 90°.
A triangle can never have two right angles or two obtuse angles — either case would already reach or exceed 180° before the third angle is counted.
Note
Two angle facts that often help
Triangle problems frequently involve crossing or parallel lines, so these two are worth keeping in mind:
Vertical angles — when two lines cross, the angles opposite each other are always equal.
Corresponding angles — when a line cuts two parallel lines, angles in the same position are equal.
Combined with the 180° rule, these let you work out angles that are not marked on the diagram at all.
Example
Working backwards in a right triangle
A right triangle has one angle of 35°. Find the remaining angle.
A right triangle already contains an angle of 90°.
The other two must therefore total 
.
⟹ 55°
In any right triangle the two non-right angles always add to 90°, which turns a three-angle problem into a single subtraction.
Note
Mistakes to avoid
Using 360° instead of 180° — that total belongs to a quadrilateral.
Assuming a triangle can contain two right angles.
Adding all three given angles when one of them is the unknown.
Forgetting that a right triangle's other two angles must sum to 90°.
Trusting how a diagram looks instead of calculating — figures are rarely drawn to scale.
Summary
- The three angles of any triangle add up to 180°, whatever its size or shape.
- Knowing two angles is always enough to find the third by subtraction.
- A right triangle has one 90° angle, so its other two angles total 90°.
- An acute triangle has all angles below 90°; an obtuse triangle has exactly one above 90°.
- Vertical and corresponding angles help find values that the diagram does not label.