Angles in a Triangle

Angle Sum in a Triangle

The three interior angles of any triangle always add up to 180°.

Angle Sum

\alpha + \beta + \gamma = 180°

Two angles of a triangle are 60° and 80°. Find the third.

$\gamma = 180° - 60° - 80° = 40°$

Types of Triangles by Angles

Type	Property
Acute	All angles less than 90°
Right	One angle equals exactly 90°
Obtuse	One angle greater than 90°
Equilateral	All angles equal 60°

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Exterior Angle

\text{ext} = \beta + \gamma

Real world: triangulation in navigation

A sailor sees a lighthouse and measures two angles from different points along the coast — say 48° and 67°. The third angle of the triangle is $180° - 48° - 67° = 65°$ . Combined with the known baseline distance, this fixes the lighthouse's exact position.

Real world: roof truss angles

A symmetric roof truss has a peak angle of 40°. Since the truss is isosceles, the two base angles are each $\frac{180° - 40°}{2} = 70°$ . A carpenter cuts the rafters at 70° to fit perfectly.

Real world: tent cross-section

A tent pole creates a triangle: the ground is the base, two guy ropes form the sides. If the pole makes a 90° angle with the ground and a rope makes a 55° angle, the third angle at the top is $180° - 90° - 55° = 35°$ .

Practice Problems

A triangle has angles 45° and 75°. Find the third angle and the exterior angle at the third vertex.

Show answer

Third angle = 180 – 45 – 75 = 60°. Exterior angle = 45 + 75 = 120°.

An isosceles triangle has one angle of 50°. What are the other two angles?

Show answer

The two base angles are equal. 50 + 2x = 180, so x = 65°. The other two angles are each 65°.