Exterior Angle Theorem
In short: Exterior angle theorem: an angle formed outside a triangle by extending one side is exactly equal to the total of the two inside angles that are not beside it.
8th GradeGeometry8.G.A.5
What is Exterior Angle Theorem?
Push one side of a triangle straight past a corner, and a new angle opens up outside the shape, and the exterior angle theorem says this new angle equals the two faraway inside angles added together, known as the remote interior angles.
With inside angles \(A\), \(B\), \(C\) and the outside angle at \(B\), the claim is that it equals \(A + C\). Reason: the outside angle and the inside angle at \(B\) form a straight line of \(180^{\circ}\), and \(A + B + C = 180^{\circ}\) too.
How to spot it
- A side of a triangle is prolonged by a ray and the angle outside the triangle carries a label.
- Two inside angles are known, and the outside angle is wanted without finding the third.
- Expressions like \(5x + 9\) label the outside angle, with two more expressions inside.
- You want a shortcut that avoids the full 180° interior sum.
The method
- Locate the outside angle and note which vertex it belongs to.
- Mark the two inside angles at the other vertices: the remote ones.
- Write the outside angle equal to the sum of those two remote angles.
- Solve, then check that the outside angle plus its neighbor inside gives \(180^{\circ}\).
Exterior Angle Theorem: solved example
Problem
A triangular roof truss has an outside angle of \((5x + 9)^{\circ}\), while the two remote inside angles are \((2x + 30)^{\circ}\) and \((x + 21)^{\circ}\). Find the outside angle and the inside angle beside it.
Solution
Apply the exterior angle theorem: \(5x + 9 = (2x + 30) + (x + 21) = 3x + 51\).
Then \(2x = 42\), so \(x = 21\). The outside angle is \(5(21) + 9 = 114^{\circ}\), and the inside neighbor is \(180 – 114 = 66^{\circ}\). Answer: \(114^{\circ}\) outside and \(66^{\circ}\) inside.
Try it
Question 1
The outside angle at one vertex is \(133^{\circ}\) and a remote inside angle is \(58^{\circ}\). What is the other remote angle?
Show the answer
\(133 – 58 = 75\), so \(75^{\circ}\).
Question 2
A triangle has inside angles \(40^{\circ}\), \(65^{\circ}\) and \(75^{\circ}\). How large is the outside angle at the \(75^{\circ}\) corner?
Show the answer
\(40 + 65 = 105\), so \(105^{\circ}\). Check: \(180 – 75 = 105\).
Exterior Angle Theorem vs. Triangle Angle Sum Theorem
The angle sum rule never leaves the triangle: three inside angles, \(180^{\circ}\). The rule here ties an outside angle to two of the inside ones. A problem that extends a side calls for this one; a problem listing only inside corners calls for the sum.
Keep going
Lessons
Worksheets and quizzes
- Angles in Triangles and Parallel Lines – Worksheet
- Angles in Triangles and Parallel Lines – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
- Triangle Angle Sum Theorem
- Supplementary Angles
- Adjacent Angles
- Isosceles Triangle Theorem
- Transversal
Whole dictionary: A–Z · 8th Grade · Geometry