Question

1. if ∠def is a straight angle, m∠deg=(23x - 3)°, and m∠gef=(12x + 8)°. find each measure.

x =

m∠deg =

m∠gef =

m∠def = 180°

Explanation:

Step1: Set up the equation

Since $\angle DEF$ is a straight - angle, $\angle DEG+\angle GEF = \angle DEF=180^{\circ}$. Given $\angle DEG=(23x - 3)^{\circ}$ and $\angle GEF=(12x + 8)^{\circ}$, we have the equation $(23x - 3)+(12x + 8)=180$.

Step2: Combine like terms

Combine the $x$ - terms and the constant terms on the left - hand side of the equation: $(23x+12x)+(-3 + 8)=180$, which simplifies to $35x+5 = 180$.

Step3: Solve for $x$

Subtract 5 from both sides of the equation: $35x=180 - 5=175$. Then divide both sides by 35: $x=\frac{175}{35}=5$.

Step4: Find $\angle DEG$

Substitute $x = 5$ into the expression for $\angle DEG$: $\angle DEG=(23x - 3)^{\circ}=(23\times5-3)^{\circ}=(115 - 3)^{\circ}=112^{\circ}$.

Step5: Find $\angle GEF$

Substitute $x = 5$ into the expression for $\angle GEF$: $\angle GEF=(12x + 8)^{\circ}=(12\times5+8)^{\circ}=(60 + 8)^{\circ}=68^{\circ}$.

Answer:

$x = 5$, $m\angle DEG=112^{\circ}$, $m\angle GEF = 68^{\circ}$