Question

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

Explanation:

Step1: Use angle - addition property

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

Step2: Combine like - terms

Combine the $x$ terms and the constant terms: $(23x+12x)+(- 3 + 8)=180$, 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, so $x=\frac{175}{35}=5$.

Step4: Find $m\angle DEG$

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

Step5: Find $m\angle GEF$

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

Step6: Recall the measure of $\angle DEF$

Since $\angle DEF$ is a straight angle, $m\angle DEF = 180^{\circ}$.

Answer:

$x = 5$
$m\angle DEG=112^{\circ}$
$m\angle GEF=68^{\circ}$
$m\angle DEF=180^{\circ}$