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Problem 270

Geometry Difficulty 2.1 Multiple choice CEMC Cayley · Canada · 2024

In $\triangle ABC$ , points
$E$ and $F$ are on $AB$ and $BC$ , respectively, such that $AE=BF$ and $BE=CF$ .

If $\angle BAC = 70\degree$ , the
measure of $\angle ABC$ is

Pick one

Official solution

Since $AE = BF$ and $BE = CF$ , then $AB = AE + BE = BF + CF = BC$ .

Therefore, $\triangle ABC$ is
isosceles with $\$\angle$ BAC = $\angle$ BCA =
$70\degree\$.$

Since the sum of the angles in $\$\triangle$
ABC $is$ $180\degree\$,$ then
$\angle ABC = 180\degree - \angle BAC - \angle BCA = 180\degree - 70\degree - 70\degree = 40\degree$

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.