Questions: Triangle Angle Sum Theorem

Question 1

Two angles of a triangle measure 47° and 85°. What is the measure of the third angle?

• A. 132°
• B. 48°
• C. 58°
• D. 38°

Answer: (B)

The three angles must sum to 180°. So the third angle is 180° − 47° − 85° = 48°. A common error is computing 47 + 85 = 132 and stopping there — that is the sum of the two known angles, not the unknown one. Another error is subtracting from 360° instead of 180°.

Question 2

The triangle angle sum theorem is a universal geometric truth that holds in most geometries, not just Euclidean geometry.

• T. True
• F. False

Answer: False

The theorem depends on Euclid's parallel postulate. In spherical geometry (like the surface of a globe), the angles of a triangle sum to more than 180° — in fact, a triangle formed by two meridians and the equator can have three 90° angles, summing to 270°. In hyperbolic geometry, they sum to less than 180°. The theorem is specifically a consequence of Euclidean geometry's axioms.

Question 3

Describe the key step in the standard proof that the interior angles of a triangle sum to 180°.

Model answer: Draw a line through one vertex parallel to the opposite side. The two base angles of the triangle equal the alternate interior angles formed with the parallel line. Together with the vertex angle, these three angles form a straight line (180°).

This proof uses alternate interior angles (which you proved hold when two parallel lines are cut by a transversal) plus the fact that a straight line measures 180°. It shows the theorem is a consequence of the parallel postulate — not a stand-alone fact — which is why it fails in non-Euclidean geometries where alternate interior angles behave differently.