204. Sides & Angles: The Great Match · bigger side always faces bigger angle

Try to break it

Drag C around. The three side lengths and three angles all shift. The longest side is always opposite the largest angle, and equal sides always face equal angles. Try to find a position where a long side faces a small angle — impossible.

How you build it

Construct a scalene triangle.

• Place point A — first vertex of the triangle.
• Place point B.
• Place point C so the three sides have visibly different lengths (a scalene triangle).
• Draw segment AB.
• Draw segment BC.
• Draw segment CA to close the triangle — the longest side will always face the largest angle and the shortest side the smallest. Drag in Break to verify.

The proof, step by step

Prove that the larger angle of a triangle lies opposite the longer side.

• Consider triangle ABC with sides AB, BC, and AC of different lengths.
• Assume without loss of generality that AC > BC > AB.
• The angle opposite AC is ∠B, opposite BC is ∠A, and opposite AB is ∠C.
• By the property of triangles, the greater side subtends the greater angle.
• Therefore, ∠B > ∠A > ∠C, proving the relationship.

Worked example

In ΔPQR, PQ = 8 cm, QR = 10 cm, and PR = 6 cm. Which angle is the largest?

The largest angle lies opposite the longest side. Here, QR = 10 cm is the longest side, so the angle opposite to it, ∠P, must be the largest angle in the triangle.

• ∠P — correct
• ∠Q
• ∠R
• All angles are equal