Question

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

Explanation:

Step1: Set up equation

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

Step2: Combine like terms

$23x+12x-3 + 8=180$, which simplifies to $35x+5 = 180$.

Step3: Isolate the variable term

Subtract 5 from both sides: $35x=180 - 5$, so $35x=175$.

Step4: Solve for x

Divide both sides by 35: $x=\frac{175}{35}=5$.

Step5: Find $m\angle DEG$

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

Step6: Find $m\angle GEF$

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

Step7: Confirm $m\angle DEF$

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

Answer:

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