CEMC Cayley, 2023

Geometry Difficulty 2.2 Junior Find the answer Canada

In the diagram, points $B$ ,
$C$ and $D$ lie on a line. Also, $\angle ABC = 90\degree$ and $\angle ACD = 150\degree$ .

The value of $x$ is

Pick one

Solution

Solution 1

Since $\angle BCD$ is a straight
angle, then $\angle ACB = 180\degree - \angle ACD = 180\degree - 150\degree = 30\degree$ .

Since the measures of the angles in $\triangle ABC$ add to $180\degree$ , then $x\degree + 90\degree + 30\degree = 180\degree$ and so $x = 180 - 120 = 60$ .

Solution 2

Since the measures of the angles in $\triangle ABC$ add to $180\degree$ , then $\angle ACB = 180\degree - \angle ABC - \angle BAC = 180\degree - 90\degree - x\degree = 90\degree - x\degree$
Since $\angle BCD = 180\degree$ , then
$\begin{align*} \angle ACB + \angle ACD & = 180\degree \\ (90\degree - x\degree) + 150\degree & = 180\degree \\ 240 - x & = 180\end{align*}$ and so $x = 60$ .

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