Theorem

Base angles of an isosceles triangle

Geometry • Triangle • Angles of a Triangle

Formula

A B = A C ⟺ ∠ A B C = ∠ A C B

In a triangle,

A B = A C

if and only if

∠ A B C = ∠ A C B

: equal sides face equal angles, and equal angles face equal sides.

How to use it

1Spot two equal sides and immediately mark the two angles opposite them equal; then finish with the angle sum.
2Run it backwards: two equal angles in a triangle prove the two sides opposite them equal, which is how an isosceles triangle is recognised from an angle condition.
3In an equilateral triangle both directions give all three angles $6 0^{\circ}$ .

Worked example

In triangle

A B C

∠ B = ∠ C = 6 5^{\circ}

. What can be said about the sides, and what is

∠ A

Step-by-step

1
$∠ B = ∠ C \Rightarrow A B = A C$
Why: Equal angles force the sides opposite them to be equal.
2
$∠ A = 18 0^{\circ} - 6 5^{\circ} - 6 5^{\circ} = 5 0^{\circ}$

Answer

A B = A C

and

∠ A = 5 0^{\circ}

Watch out

The equal angles are the ones OPPOSITE the equal sides, not the pair enclosing them. The converse half — equal angles giving equal sides — is the direction most often forgotten, and it is the one that turns an angle condition into a length.

Builds on

Angle sum of a triangle

Also known as

isosceles triangle base angles, pons asinorum, equal sides equal angles, angles opposite equal sides, converse of the isosceles triangle theorem

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (I)

2019 (II)

2019 (I)

2018 (II)

2018 (I)

2017 (II)

2017 (I)