Chapter 3.2, Problem 45HP

To determine

To find: The value of $x$ and $y$ .

Answer to Problem 45HP

The two possible solutions are: $y=3$ , $x=171$ and $y=5$ , $x=155$.

Explanation of Solution

Given information:

The given information

Calculation:

${\left(y^2\right)}^{\circ }$ and ${\left(8y-15\right)}^{\circ }$ alternate interior angles so by the Alternate Interior Angles Theorem, they are congruent:

$\begin{array}{c}{\left(y^2\right)}^{\circ }\cong {\left(8y-15\right)}^{\circ }\\ y^2=8y-15\\ y^2-8y+15=0\end{array}$

$\begin{array}{c}{y}^2-8y+15=0\\ y^2-5y-3y+15=0\\ \left(y-5\right)\left(y-3\right)=0\end{array}$

Now,

$y=3,5$

${\left(x\right)}^{\circ }$ and ${\left(y^2\right)}^{\circ }$ form a linear pair so they are supplementary:

$\begin{array}{c}{\left(x\right)}^{\circ }+{\left(y^2\right)}^{\circ }\cong {180}^{\circ }\\ x+y^2={180}^{\circ }\end{array}$

When $y=3$ ,

$\begin{array}{c}x+3^2={180}^{\circ }\\ x=171\end{array}$

When $y=5$ ,

$\begin{array}{c}x+5^2={180}^{\circ }\\ x=155\end{array}$

So, there are two possible solutions are: $y=3$ , $x=171$ and $y=5$ , $x=155$ .

Therefore, the two possible solutions are: $y=3$ , $x=171$ and $y=5$ , $x=155$.