Question

Find the value of x and y in the given figure.

Answer 1

Hint: We will use the angle sum property to find the angles x and y in the given figure. The sum of the angles of a triangle is 180o. Also, the altitude through a vertex in a triangle is perpendicular to the opposite side.

Complete step-by-step solution -
We will consider the angle sum property in the two triangles ACB and DCB which are right-angled triangles at C.
$ In\;\vartriangle ACB, \\ $
$\Rightarrow \angle {\text{A}} + \angle {\text{C}} + \angle {\text{B}} = {180^{\text{o}}} \\ $
 $\Rightarrow {40^{\text{o}}} + {90^{\text{o}}} + {{\text{x}}^{\text{o}}} = {180^{\text{o}}} \\ $
$\Rightarrow {{\text{x}}^{\text{o}}}\; = \;{\left( {180\; - \;90\; - 40} \right)^{\text{o}}} \\$
$\Rightarrow {{\text{x}}^{\text{o}}} = {50^{\text{o}}} \\ $
$ In\;\vartriangle DCB, \\ $
$\Rightarrow \angle {\text{D}} + \angle {\text{C}} + \angle {\text{B}} = {180^{\text{o}}} \\ $
$\Rightarrow {60^{\text{o}}} + {90^{\text{o}}} + {{\text{y}}^{\text{o}}} = {180^{\text{o}}} \\ $
$\Rightarrow {{\text{y}}^{\text{o}}} = {\left( {180 - 90 - 60} \right)^{\text{o}}} \\ $
$\Rightarrow {{\text{y}}^{\text{o}}} = {30^{\text{o}}} \\ $
This is the required answer.

Note: We can also verify the answer by applying the angle sum property in the bigger triangle, which is ABD. Applying the angle sum property,
$\begin{gathered}
  \angle {\text{A}} + \angle {\text{B}} + \angle {\text{D}} = {180^{\text{o}}} \\
  40 + \left( {{\text{x}} + {\text{y}}} \right) + 60 = {180^{\text{o}}} \\
  40 + \left( {50 + 30} \right) + 60 = {180^{\text{o}}} \\
  {180^{\text{o}}} = {180^{\text{o}}} \\
\end{gathered} $
Hence, the answer is verified.