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: Use vertical - angle property

Since $\angle DEG$ and $\angle GEF$ are a linear - pair of angles, $m\angle DEG + m\angle GEF=180^{\circ}$. 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, so $x=\frac{192}{12}=16$.

Step4: Find $m\angle GEF$

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

Step5: Assume $\angle DEH$ and $\angle GEF$ are vertical angles

If $\angle DEH$ and $\angle GEF$ are vertical angles, then $m\angle DEH=m\angle GEF$. So, $9y + 5=104$.

Step6: Solve for $y$

Subtract 5 from both sides: $9y=104 - 5 = 99$. Then divide both sides by 9, so $y=\frac{99}{9}=11$.

Answer:

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