CEMC Pascal, 2023

Geometry Difficulty 2.6 Junior Find the answer Canada

In the diagram, $A$ , $B$ , $D$ , $F$ , and $G$ lie on a vertical line, $\triangle BCD$ is right-angled at $C$ , and $\triangle DEF$ is right-angled at $E$ . Also, $\angle ABC = x\degree$ , $\angle CDE = 80\degree$ , and $\angle EFG = y\degree$ .

What is the value of $x+y$ ?

Pick one

Solution

Since $\angle ABD = 180\degree$ and $\angle ABC = x\degree$ , then $\angle CBD = 180\degree - x\degree$ .

Since the measures of the angles in $\triangle BCD$ add to $180\degree$ , then $\angle BDC = 180\degree - (180\degree - x\degree) - 90\degree = x\degree - 90\degree$ Similarly, $\angle GFD = 180\degree$ and $\angle FDE = y\degree - 90\degree$ .
Finally, $\angle BDF = 180\degree$
and so $\begin{align*} \angle BDC + \angle CDE + \angle FDE & = 180\degree \\ (x\degree - 90\degree) + 80\degree + (y\degree - 90\degree) & = 180\degree \\ x + y - 100 & = 180\end{align*}$ and so $x + y = 280$ .

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