CEMC Cayley, 2024

Geometry Difficulty 2.1 Junior Find the answer Canada

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

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
$\angle ABC = 180\degree - \angle BAC - \angle BCA = 180\degree - 70\degree - 70\degree = 40\degree$

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