The three angles of a triangle always add up to 180°
You already know this — but parallel lines cut by two transversals prove it! Draw a line through the top vertex parallel to the base. The two new angles match the base angles (alternate interior angles), and together with the top angle they form a straight line: 180°.
1
Spot the parallel lines — the two transversals form the sides of the triangle.
Alternate interior angles are EQUAL
2
Match angles — each angle on the top line equals a base angle of the triangle.
3
Add to 180° — the angles across the top form a straight line.
x = 180° − (other two angles)
Exterior Angle Theorem
Exterior angle = sum of the two REMOTE interior angles
Extend one side of the triangle. The angle formed outside equals the two interior angles that are not adjacent (not next-door) to it.
1
Find the exterior angle — it sits on the extended side, outside the triangle.
2
Find the two remote interior angles — the two angles far away from it.
3
Set up the equation:
exterior = remote + remote
Worked Example
A triangle has interior angles of 40° and 65°. One side is extended to form an exterior angle next to the third angle. Find the exterior angle.
1
The remote interior angles are 40° and 65°
2
Exterior angle = 40° + 65° = 105°
3
Check with a linear pair: the third interior angle is 180° − 105° = 75°, and 40 + 65 + 75 = 180 ✔
