Question

Using information given in the adjacent figure, find the values of $x$ and $y$.

Answer 1

Side $BC$ of $△ABC$ has been produced to $D$.
$$\begin{array}{cc}& \text{Exterior}\angle ACD=\angle ABC+\angle BAC\\ & \therefore {100}^{\circ }={65}^{\circ }+x^{\circ }\\ & \Rightarrow x^{\circ }=({100}^{\circ }-{65}^{\circ })={35}^{\circ }\\ & \therefore \angle CAD=\angle BAC={35}^{\circ }\end{array}$$
In $△ACD$, we have :
$$\begin{array}{cc}& \angle CAD+\angle ACD+\angle CDA={180}^{\circ }\text{(Angle sum property of triangle)}\\ & \Rightarrow {35}^{\circ }+{100}^{\circ }+y^{\circ }={180}^{\circ }\\ & \Rightarrow {135}^{\circ }+y^{\circ }={180}^{\circ }\\ & \Rightarrow y^{\circ }=({180}^{\circ }-{135}^{\circ })={45}^{\circ }\\ & \text{Hence,}x={35}^{\circ },y={45}^{\circ }\end{array}$$