An exterior angle equals the sum of the two remote interior angles.
Key formulas
Why it works
Interior angles sum to 180°. The adjacent interior and exterior also sum to 180°. Eliminate the adjacent interior to leave the exterior as the sum of the other two.
Example
Remote interiors 40° and 65° → exterior 105°.
Diagram habit
Extend one side clearly. Students sometimes use an adjacent angle instead of the remote pair.
Worked example
Worked example
Problem. In a triangle, two remote interior angles are 48° and 65°. Find the exterior angle at the third vertex.
Solution. Exterior angle = 48° + 65° = 113°.
Check
Adjacent interior would be 75° if exterior is 105°; 75+40+65=180.
Review quiz
Remote interiors 30° and 55°. Exterior angle is:
Practice questions
- Remote interiors 28° and 47°. Find the exterior.
- Draw a triangle and mark one exterior angle with its remote interiors.
Study notes
Is the exterior unique?
Each vertex has two exterior directions that are vertically opposite equal angles.
More in Plane Geometry
- Circle tangents: the radius is perpendicular
- Right-triangle lengths: Pythagoras and altitude
- Similar triangles: matching sides in proportion
- Chord theorem: perpendicular from the centre
- Isosceles triangles and equal base angles
- Alternate segment theorem with a short proof
- Converse of Pythagoras: proving a right angle
- Heron’s formula when height is hidden