Triangle Angle Sum Theorem

Geometry · Axiom Academy

LESSON Triangle Angle Sum Theorem The interior angles of any triangle always add up to 180°. Take any triangle and call its three interior angles A , B , and C . Tear off the three corners and lay them side by side around a single point — they snap together into a perfectly straight line. A straight line is 180° , so the three angles must sum to 180°. The sum of the interior angles of any triangle is always 180°. 2. Why It Works (Visual Proof) Here's a clean way to see why the angles always sum to 180°. Draw a line through the top vertex parallel to the base . The triangle's two slanted sides cut that line, creating alternate interior angles equal to B and C. Along the straight line at the top, those copies plus A fill a straight angle. Label the three interior angles A, B, and C. Through the top vertex, draw a line parallel to the base. The two sides act as transversals. The angle B′ on the line equals B, and C′ equals C — they are alternate interior angles. At the top vertex, B′ + A + C′ lie along a straight line, so they sum to 180°. Since B′ = B and C′ = C, replacing them in B′ + A + C′ = 180° gives A + B + C = 180° — the theorem, proven. 3. Using It to Find a Missing Angle The theorem turns into a one-line recipe. If you know two angles, the third is forced — it's exactly whatever makes the total reach 180°. Find the missing angle in this right triangle. All three angles are equal. Find x. 3x = 180°, so x = 60° (an equilateral triangle).

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