DWG NO. 04.2 — Lesson 2 of 6

Triangle Angle Sum and Exterior Angle Theorems

Unit 4: Congruent Triangles · ~15–30 min

Objective

Apply the Triangle Angle Sum Theorem and the Exterior Angle Theorem to find missing angle measures.

The Triangle Angle Sum Theorem

The three interior angles of any triangle — no matter how stretched or skewed — always add up to the same total.

Triangle Angle Sum Theorem: The measures of the interior angles of a triangle sum to 180°.

This means that if you know two angles of a triangle, the third is automatically determined: subtract the two known angles from 180°.

Exterior angles

Extend one side of a triangle past a vertex, and the angle formed outside the triangle is an exterior angle. Each exterior angle has two remote interior angles — the two angles of the triangle that aren't next to it.

Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the two remote interior angles.
An exterior angle and its adjacent interior angle are supplementary (they form a straight line), since they sit on a line together.

Extending side AC past C creates exterior angle BCD, equal to the sum of remote interior angles A and B.

Worked Example 1 · Find the missing interior angle

ProblemTriangle has angles 48° and 77°. Find the third angle.

1.By the Angle Sum Theorem, all three add to 180°: 48 + 77 + x = 180.

2.125 + x = 180, so x = 55.

Third angle = 55°

Worked Example 2 · Use the Exterior Angle Theorem

ProblemIn triangle ABC, angle A = 40° and angle B = 65°. Find the exterior angle at C.

1.The exterior angle at C equals the sum of the two remote interior angles, A and B.

2.Exterior angle = 40 + 65 = 105°.

3.Check: interior angle C = 180 − 105 = 75°, and 40 + 65 + 75 = 180. ✓

Exterior angle at C = 105°

Guided practice

1.A triangle has angles 90° and 34°. Find the third angle.
Show answer
180 − 90 − 34 = 56°.
2.An exterior angle of a triangle measures 118°. One remote interior angle is 50°. Find the other remote interior angle.
Show answer
118 = 50 + x, so x = 68°.
3.Can a triangle have interior angles of 90°, 90°, and 0°? Explain using the Angle Sum Theorem.
Show answer
No — while 90 + 90 + 0 = 180 numerically, an angle can't measure 0° in an actual triangle (that would collapse a side to zero length). A valid triangle also can't have two 90° angles, since the sides forming both right angles would never meet to close the triangle.
4.The interior angle adjacent to an exterior angle measures 72°. What does the exterior angle measure?
Show answer
An exterior angle and its adjacent interior angle are supplementary: 180 − 72 = 108°.
5.Triangle DEF has angle D = 3x, angle E = 2x, and angle F = x. Find x and each angle.
Show answer
3x + 2x + x = 180, so 6x = 180 and x = 30. Angles: D = 90°, E = 60°, F = 30°.

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