222. The Greater Angle's Side · If angles differ, opposite sides follow suit

Try to break it

Drag A, B, or C to reshape △ABC; drag D along BC. Whenever ∠A > ∠B, the side BC (opposite ∠A) is longer than side AC (opposite ∠B). Try to find a position where ∠A > ∠B yet BC < AC — impossible. Bigger angles always face longer sides.

How you build it

Construct a triangle with an equal-angle point.

• Place point A as the first vertex.
• Place point B to the right of A, level with it.
• Place point C above AB, nearer to B than to A, so ∠A > ∠B.
• Draw the segment from A to B.
• Draw the segment from B to C.
• Draw the segment from C to A.

The proof, step by step

Prove that the side opposite the greater angle is the longer side.

• In ΔABD, ∠BAD = ∠B (by construction). Therefore, AD = BD. (Sides opposite to equal angles are equal.)
• In ΔADC, AD + DC > AC. (Sum of any two sides of a triangle is always greater than the third side.)
• Substitute AD = BD into the inequality: BD + DC > AC.
• Since BD + DC = BC, we get BC > AC. Hence Proved.

Worked example

In ΔPQR, ∠P = 80° and ∠Q = 50°. Which side of the triangle is the longest?

∠R = 180° - (80° + 50°) = 50°. Since ∠P (80°) > ∠Q (50°) = ∠R (50°), the side opposite ∠P is the longest. Side opposite ∠P is QR. Hence, QR is the longest side.

• PQ
• QR — correct
• PR
• Cannot be determined