Exterior Angle Theorem
An exterior angle equals the sum of the two remote interior angles.
About this topic
An exterior angle of a triangle is formed by extending one side. The theorem says it equals the sum of the two remote interior angles, those not adjacent to it. If the remote angles are 50° and 60°, the exterior angle is 110°.
It follows from the triangle angle sum, since the exterior angle and its adjacent interior angle form a straight line. The theorem gives a shortcut for finding angles and appears in many proofs.
Worked example: a triangle has angles 50° and 60° at two vertices. Extend a side at the third vertex, whose interior angle is 70°. The exterior angle is 180° − 70° = 110°, which equals 50° + 60°, matching the theorem. If an exterior angle is 125° and one remote interior angle is 80°, the other remote interior angle is 45°. This saves the extra step of finding the third interior angle.
The theorem is used in surveying and in proofs about parallel lines and inequalities, such as showing an exterior angle is larger than either remote angle. A common mistake is using the adjacent interior angle instead of the remote ones, which gives a wrong sum. Always mark the two angles far from the exterior angle first. Also, each vertex has two exterior angles, equal to each other as vertical angles, and the exterior angles of a triangle total 360°.
Key ideas
- Extend one side of the triangle
- Exterior + adjacent interior = 180°
- Handy shortcut for missing angles
Quick check
Use cases: where Exterior Angle Theorem helps
- Homework: review the key idea behind Exterior Angle Theorem before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Exterior Angle Theorem without notes.
- Visual learning: play with the interactive lab to see how Exterior Angle Theorem changes when you adjust its inputs.
- Teaching a friend: explain Exterior Angle Theorem in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Exterior Angle Theorem?
An exterior angle equals the sum of the two remote interior angles.
What will I learn in Exterior Angle Theorem?
Extend one side of the triangle. Exterior + adjacent interior = 180°. Handy shortcut for missing angles.
Is Exterior Angle Theorem part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Exterior Angle Theorem lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Exterior Angle Theorem?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.