Question

1. if m∠deg=(5x - 4)°, m∠gef=(7x - 8)°, m∠deh=(9y + 5)°, find the values of x and y.

Explanation:

Step1: Use vertical - angle property

Since $\angle DEG$ and $\angle GEF$ are a linear - pair and $\angle DEG$ and $\angle FEH$ are vertical angles, and $\angle GEF$ and $\angle DEH$ are vertical angles. Also, $\angle DEG+\angle GEF = 180^{\circ}$ (linear - pair of angles).
So, $(5x - 4)+(7x - 8)=180$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $5x+7x-4 - 8=180$, which gives $12x-12 = 180$.

Step3: Solve for $x$

Add 12 to both sides of the equation: $12x=180 + 12=192$. Then divide both sides by 12: $x=\frac{192}{12}=16$.

Step4: Use the vertical - angle relationship for $y$

Since $\angle GEF=\angle DEH$ (vertical angles), we set $7x - 8=9y + 5$. Substitute $x = 16$ into the equation: $7\times16-8=9y + 5$.
$112-8=9y + 5$, so $104=9y + 5$.

Step5: Solve for $y$

Subtract 5 from both sides: $9y=104 - 5 = 99$. Then divide both sides by 9: $y = 11$.

Answer:

$x = 16$, $y = 11$