The Exterior Angle Theorem is a powerful shortcut for finding missing angles in triangles. When one side of a triangle is extended, it creates an exterior angle outside the triangle. This exterior angle is related to the two interior angles that are not next to it.
The theorem matters because it turns many geometry problems into simple addition or subtraction.
Key Facts
- Exterior Angle Theorem: m∠ACD = m∠A + m∠B when BC is extended through C to D.
- The two remote interior angles are the two triangle angles not adjacent to the exterior angle.
- An exterior angle and its adjacent interior angle form a linear pair, so their measures add to 180°.
- Triangle angle sum: m∠A + m∠B + m∠C = 180°.
- If m∠A = 45° and m∠B = 70°, then the exterior angle at C is 45° + 70° = 115°.
- The exterior angle is always greater than either remote interior angle alone in a triangle.
Vocabulary
- Exterior angle
- An exterior angle is an angle formed outside a polygon by extending one of its sides.
- Remote interior angles
- Remote interior angles are the two angles inside a triangle that are not adjacent to the chosen exterior angle.
- Adjacent interior angle
- The adjacent interior angle is the triangle angle that shares a side and vertex with the exterior angle.
- Linear pair
- A linear pair is two adjacent angles whose nonshared sides form a straight line and whose measures add to 180°.
- Triangle angle sum
- The triangle angle sum is the rule that the three interior angles of any triangle add to 180°.
Common Mistakes to Avoid
- Adding the exterior angle to the adjacent interior angle as if they were remote angles is wrong because those two angles form a linear pair and add to 180°.
- Using all three interior angles in the theorem is wrong because the exterior angle equals only the sum of the two remote interior angles.
- Labeling the wrong angles as remote interior angles is wrong because the remote angles must be inside the triangle and not touch the exterior angle's vertex.
- Assuming the exterior angle equals the adjacent interior angle is wrong because they are supplementary, not usually equal.
Practice Questions
- 1 In triangle ABC, side BC is extended through C to D. If m∠A = 38° and m∠B = 79°, find m∠ACD.
- 2 An exterior angle of a triangle measures 132°. One remote interior angle measures 57°. Find the other remote interior angle.
- 3 A student says an exterior angle must be added to the interior angle next to it to get the sum of the two remote interior angles. Explain why this statement is incorrect using the Exterior Angle Theorem and the idea of a linear pair.