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Geometry Difficulty 1.3 Junior Find the answer Canada

In the diagram, a semi-circle has centre $O$ and diameter $AC$ . Also, $B$ is a point on the circumference such
that $\angle BAC = 25\degree$ .

The measure of $\angle BOC$ is

Pick one

Solution

Since $O$ is the centre of the
circle, then $OA = OB = OC$ .

This means that $\triangle AOB$ and
$\triangle COB$ are both isosceles
with $\$\angle$ ABO = $\angle$ BAO = $\angle$ BAC =
$25\degree\$.$

Thus, $\$\angle$ AOB = $180\degree - \angle$ ABO -
$\angle$ BAO = $130\degree\$.$

Since $\angle AOC$ is a straight
angle, then $\$\angle$ BOC = $180\degree - \angle$
AOB = $180\degree - 130\degree = 50\degree\$.$

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