The Angles of a Triangle Always Add Up to 180°

No matter what a triangle looks like — tall and narrow, flat and wide, or anywhere in between — its three interior angles will always add up to exactly 180°. That's one of the most fundamental facts in geometry, and it shows up constantly.

Finding a Missing Angle

If you know two angles of a triangle, finding the third is straightforward: subtract the two known angles from 180.

Suppose triangle ABC has ∠A = 40° and ∠B = 60°. What is ∠C?

$$\angle C = 180 - 40 - 60 = 80°$$

That's it. You can verify by adding all three: \(40 + 60 + 80 = 180\). ✓

Once you're comfortable with this, you won't need to write out the subtraction every time — you'll mentally note that two known angles sum to 100°, so the missing one must be 80°.

Equilateral Triangles

An equilateral triangle has three sides of equal length. What does that mean for the angles? Since all three angles must be equal and sum to 180°, each one must be exactly 60°. You can verify with the equation:

$$x + x + x = 180$$ $$3x = 180$$ $$x = 60$$

Angles in a Ratio

Sometimes a problem gives you angles expressed as a ratio rather than as specific values. The approach is to assign a variable and use the fact that the sum must be 180°.

Example: The angles of a triangle are in the ratio 4 : 5 : 9. Find the measure of each angle.

Let the angles be \(4x\), \(5x\), and \(9x\). Their sum must equal 180°:

$$4x + 5x + 9x = 180$$ $$18x = 180$$ $$x = 10$$

So the three angles are \(4(10) = 40°\), \(5(10) = 50°\), and \(9(10) = 90°\).

Notice the angles add to \(40 + 50 + 90 = 180\). ✓ And the smallest angle — 40° — corresponds to the smallest number in the ratio (4), which makes sense.

A Note on Exterior Angles

Each angle discussed above is an interior angle — measured inside the triangle. If you extend one side of the triangle outward, the angle formed on the outside is called an exterior angle, and it has its own useful properties.