Question

1. if (mangle deg=(5x - 4)^{circ}), (mangle gef=(7x - 8)^{circ}) and (mangle deh=(9y + 5)^{circ})

a. find the value of x
b. find (mangle gef)
c. find the value of y.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle DEG$ and $\angle GEF$ are adjacent and form a linear - pair, and $\angle DEG$ and $\angle HEF$ are vertical angles, $\angle GEF$ and $\angle DEH$ are vertical angles. Let's assume $\angle DEG=\angle HEF$ and $\angle GEF = \angle DEH$. Since $\angle DEG=(5x - 4)^{\circ}$ and $\angle GEF=(7x - 8)^{\circ}$, and $\angle DEG+\angle GEF = 180^{\circ}$ (linear - pair of angles), we have the equation $(5x - 4)+(7x - 8)=180$.

Step2: Simplify the equation

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

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: Find the measure of $\angle GEF$

Substitute $x = 16$ into the expression for $\angle GEF$. $\angle GEF=(7x - 8)^{\circ}$. So $\angle GEF=7\times16-8=112 - 8=104^{\circ}$.

Step5: Find the value of y

Since $\angle GEF=\angle DEH$ and $\angle DEH=(9y + 5)^{\circ}$, and $\angle GEF = 104^{\circ}$, we set up the equation $9y+5=104$. Subtract 5 from both sides: $9y=104 - 5=99$. Divide both sides by 9: $y = 11$.

Answer:

A. $x = 16$
B. $m\angle GEF=104^{\circ}$
C. $y = 11$