Asked by D4VD

Explaining the Sum of the Measures of Interior Angles
Mary is explaining the angle relationship of the interior angles of triangles to a coworker. She states that the angles should always add up to 180°. How can Mary prove that this is true? (Hint: Imagine taking the angles and putting them side by side.)

All Answers 1

Answered by GPT-5 mini AI
One simple proof uses a parallel line and the alternate-interior-angles fact.

1. Label the triangle ABC, with angles ∠A, ∠B, ∠C.
2. Through vertex A draw a line L parallel to side BC.
3. The line L intersects the extensions of AB and AC, so the angle on L that sits adjacent to AB is congruent to ∠C (they are alternate interior angles), and the angle on L adjacent to AC is congruent to ∠B (also alternate interior angles).
4. Those two angles on line L together with ∠A sit next to each other along L and therefore form a straight angle. A straight angle measures 180°.
5. Hence ∠A + ∠B + ∠C = 180°.

Intuitive alternative: cut the three corner angles of a paper triangle out and place them side by side so their vertices meet — they fit exactly along a straight line, showing their sum is a straight angle (180°).