Theorem

The greater side faces the greater angle

Geometry • Triangle • Triangle Inequality

Statement

In any triangle, of two unequal angles the greater one faces the longer side; and of two unequal sides the longer one faces the greater angle. So in

△ A B C

∠ A > ∠ B

if and only if

B C > C A

. Equal angles face equal sides, which is the isosceles case.

How to use it

1To compare two sides, compare the angles opposite them — no lengths are needed.
2In a right or obtuse triangle the largest angle gives the longest side at once: the hypotenuse, or the side facing the obtuse angle.
3Combine with the angle sum to settle a comparison when only two angles are given.
4In a sub-triangle cut off by a cevian, an exterior angle is larger than either remote interior angle, which often supplies the strict inequality needed.

Worked example

△ A B C

∠ A = 8 0^{\circ}

∠ B = 6 0^{\circ}

and

∠ C = 4 0^{\circ}

. Arrange the sides in increasing order.

Step-by-step

1
$∠ C < ∠ B < ∠ A$
Why: Order the three given angles.
2
$A B < C A < B C$
Why: Each side faces the angle of the same rank: $A B$ faces $∠ C$ , $C A$ faces $∠ B$ , $B C$ faces $∠ A$ .

Answer

A B < C A < B C

Watch out

The correspondence is side-to-OPPOSITE-angle, not side-to-adjacent-angle; naming which side faces which angle is where this goes wrong. It orders sides but says nothing about how much longer one is — that needs the sine rule or Pythagoras.

Builds on

Angle sum of a triangle

Also known as

greater angle opposite greater side, longest side faces largest angle, side angle inequality, triangle side ordering

Appears in past papers

2020 (I)

Q82