Chapter 6.4, Problem 45HP

To determine

To calculate:

The values of $\mathrm{x}$ and $\mathrm{y}$

Answer to Problem 45HP

The values of $\mathrm{x}$ and $\mathrm{y}$ are $6$ and $-10$ respectively.

Explanation of Solution

Given information:

  

  $\begin{array}{l}\mathrm{m}\angle \mathrm{E}\mathrm{A}\mathrm{B}=4\mathrm{x}+6\\ \mathrm{m}\angle \mathrm{D}\mathrm{E}\mathrm{C}=10-11\mathrm{y}\\ \mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{C}=60°\end{array}$

Calculation:

It is given that a rectangle $▭\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ which has four right angles. So,

  $\begin{array}{l}\mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{C}+\mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{A}=90°\\ 60°+\mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{A}=90°\\ \mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{A}=30°\end{array}$

The diagonals of a rectangle are congruent and bisect each other. So, $\mathrm{\Delta }\mathrm{E}\mathrm{A}\mathrm{B}$ is an isosceles triangle with $\angle \mathrm{A}\cong \angle \mathrm{B}$

So, $\mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{A}=30°$

But from the given information:

  $\begin{array}{l}\mathrm{m}\angle \mathrm{E}\mathrm{B}\mathrm{A}=30°\\ 4\mathrm{x}+6=30\\ 4\mathrm{x}=24\\ \mathrm{x}=6\end{array}$

Know that: The sum of a triangle is $180°$ :

  $\mathrm{m}\angle \mathrm{A}\mathrm{E}\mathrm{B}=180-\left(30°+30°\right)=120°$

According to the vertical angle theorem:

  $\mathrm{m}\angle \mathrm{A}\mathrm{E}\mathrm{B}\cong \mathrm{m}\angle \mathrm{D}\mathrm{E}\mathrm{C}$

So,

  $\begin{array}{l}\mathrm{m}\angle \mathrm{D}\mathrm{E}\mathrm{C}=120°\\ 10-11\mathrm{y}=120\\ -11\mathrm{y}=110\\ \mathrm{y}=-10\end{array}$

Hence, the value of $\mathrm{x}$ and $\mathrm{y}$ are calculated as $6$ and $-10$ respectively.