Angle sum property

The most famous fact about triangles: the three angles always sum to $180^\circ$, no matter the shape or size.

Idea

The claim. For any triangle $ABC$, $\angle A + \angle B + \angle C = 180^\circ.$

A paper proof. Cut out a paper triangle. Tear off the three corners. Place them side by side along a straight edge. The three angles fit exactly along a straight line ($180^\circ$). This works for every triangle.

A line-and-angle proof. Through vertex $A$, draw a line parallel to $BC$. The three angles at $A$ , interior $\angle A$ plus the two angles between the new line and sides $AB$, $AC$ , together form a straight line ($180^\circ$). By the alternate-interior-angles rule (with $BC$ parallel to the new line), those two outer angles equal $\angle B$ and $\angle C$ respectively. So $\angle A + \angle B + \angle C = 180^\circ$. ✓

Applications.

• Find a missing angle. If two angles are $40^\circ$ and $80^\circ$, the third is $180 - 40 - 80 = 60^\circ$.
• In an equilateral triangle, each angle is $\tfrac{180^\circ}{3} = 60^\circ$.
• In a right triangle, the other two angles sum to $90^\circ$. So if one is $35^\circ$, the other is $55^\circ$.
• In an isosceles triangle with apex angle $40^\circ$, each base angle is $(180-40)/2 = 70^\circ$.

The angle sum property combined with the isosceles symmetry handles a huge fraction of triangle problems.

Worked examples

Example 1. Triangle with angles $50^\circ$ and $80^\circ$. Find the third.

$180 - 50 - 80 = 50^\circ$.

Example 2. Triangle with two angles in ratio $2:3$, and the third $80^\circ$. Find all three.

The other two angles sum to $180 - 80 = 100^\circ$. Split in ratio $2:3$: $40^\circ$ and $60^\circ$. So angles are $40^\circ, 60^\circ, 80^\circ$.

Example 3. An isosceles triangle has base angles $55^\circ$ each. Find the apex angle.

Apex $= 180 - 55 - 55 = 70^\circ$.

Example 4. A right triangle has one acute angle $32^\circ$. Find the other acute angle.

The two acute angles sum to $90^\circ$. So the other is $90 - 32 = 58^\circ$.

Try it yourself

1. Find the missing angle: $45^\circ, 65^\circ, ?$.
2. An equilateral triangle has angles of what measure?
3. Two angles of a triangle are equal and the third is $40^\circ$. Find the two equal angles.
4. A right triangle has one acute angle $48^\circ$. Find the other.
5. The three angles of a triangle are $(x-10)^\circ, (x)^\circ, (x+10)^\circ$. Find each.
6. An isosceles triangle with apex angle $90^\circ$ , find each base angle.
7. The angles are $3x, 4x, 5x$. Find $x$ and each angle.
8. Can a triangle have angles $60^\circ, 70^\circ, 80^\circ$? Justify.

Activity

Use paper triangles to do the "tear-off" proof: cut any triangle, tear off the three corners, lay them along a straight line. The corners fit exactly , that is the $180^\circ$ sum visualised.

Draw three triangles of different shapes on grid paper. Measure each angle with a protractor; check the sum is close to $180^\circ$ each time (small errors from measurement are normal).