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Problem 227

Geometry Difficulty 1.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2015

In a triangle, the measure of one of the angles is $45^{\circ}$ . The measures of the other two angles in the triangle are in the ratio $4:5$ . What is the measure of the largest angle in the triangle?

Pick one

Official solution

Since the sum of the measures of the three angles in any triangle is $180\degree$ , then the sum of the measures of the two unknown angles in the given triangle is $180\degree-45\degree=135\degree$ .

The measures of the two unknown angles are in the ratio $4:5$ , and so one of the two angle measures is $\frac{5}{4+5}=\frac59$ of the sum of the two angles, while the other angle measures $\frac{4}{4+5}=\frac49$ of the sum of the two angles.

That is, the larger of the two unknown angles measures $\frac59\times135\degree=75\degree$ , and the smaller of the unknown angles measures $\frac49\times135\degree=60\degree$ .

We may check that $60\degree+75\degree+45\degree=180\degree$ .
The largest angle in the triangle measures $75\degree$ .

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