The outside angle’s two partners
To make an exterior angle, extend one side of a triangle past a vertex. The new outside angle sits next to one interior angle and opens away from the triangle. The interior angle beside it is not one of the remote angles. The other two interior angles are the pair you add together.
Imagine a triangle whose remote interior angles measure 42° and 68°. The exterior angle formed by extending the third side measures 42° + 68° = 110°. You can check the answer another way: the inside angle next to 110° must be 70° because 110° + 70° = 180°, and 42° + 68° + 70° = 180°.
That second check shows why the theorem works. The outside angle and its neighboring inside angle make a straight angle, so they total 180°. The three inside angles of a triangle also total 180°. Subtracting the neighboring angle from each relationship leaves the outside angle equal to the two remote angles together.
This is helpful when a diagram gives an outside angle but hides one of the inside measurements. Instead of beginning with a long chain of angle facts, identify the remote pair and write one sum. The picture becomes easier to read because the outside angle is carrying information from two places inside the triangle.
A quick drawing can prevent a wrong equation. Circle the exterior angle, lightly mark the interior angle beside it, and then point to the two angles that remain. If the side has not actually been extended, you may be looking at an ordinary interior angle rather than an exterior one. The theorem depends on that extension.
- A triangle's exterior angle equals the sum of the two remote interior angles.
- The remote interior angles are the two triangle angles not touching the exterior angle.
- An exterior angle is greater than either one remote interior angle by itself because it equals their sum.
- The theorem depends on choosing the correct exterior angle at a side extension, not just any angle near the triangle.