It's important to know that a rectangle has opposite parallel equal sides, which means that the diagonal is a transversal between the parallels.
We can deduct that the angle 7y-2 and the angle 4y+13 are alternate interior angles because they are between parallels and at different sides of the transversal, which means those angles are equivalent.
[tex]7y-2=4y+13[/tex]
Let's solve for y, first, we subtract 4y from each side.
[tex]\begin{gathered} 7y-4y-2=4y-4y+13 \\ 3y-2=13 \end{gathered}[/tex]
Then, add 2 on each side.
[tex]\begin{gathered} 3y-2+2=13+2 \\ 3y=15 \end{gathered}[/tex]
At last, divide both sides by 3.
[tex]\begin{gathered} \frac{3y}{3}=\frac{15}{3} \\ y=5 \end{gathered}[/tex]
Once we have the value of y, we use it to find x. We know that the angle 5x+7 and the angle 7y-2 are complementary because if the figure is a rectangle, then all its interior angles measure 90°.
[tex]5x+7+7y-2=90[/tex]
Now we use the value of y and solve for x.
[tex]\begin{gathered} 5x+7+7\cdot5-2=90 \\ 5x+7+35-2=90 \\ 5x+40=90 \end{gathered}[/tex]
Subtract 40 from each side.
[tex]\begin{gathered} 5x+40-40=90-40 \\ 5x=50 \end{gathered}[/tex]
Divide both sides by 5.
[tex]\begin{gathered} \frac{5x}{5}=\frac{50}{5} \\ x=10 \end{gathered}[/tex]
Therefore, the value of x is 10 and the value of y is 5.
