Question

1. if $mangle def=(7x + 4)^{circ}$, $mangle deg=(5x + 1)^{circ}$, and $mangle gef = 23^{circ}$, find each measure.

Explanation:

Step1: Use angle - addition postulate

Since $\angle DEF=\angle DEG+\angle GEF$, we have the equation $(7x + 4)=(5x + 1)+23$.

Step2: Simplify the right - hand side of the equation

$(5x + 1)+23=5x+24$. So the equation becomes $7x + 4=5x+24$.

Step3: Isolate the variable terms

Subtract $5x$ from both sides: $7x-5x + 4=5x-5x+24$, which simplifies to $2x+4 = 24$.

Step4: Isolate the variable

Subtract 4 from both sides: $2x+4 - 4=24 - 4$, getting $2x=20$.

Step5: Solve for x

Divide both sides by 2: $\frac{2x}{2}=\frac{20}{2}$, so $x = 10$.

Step6: Find $m\angle DEG$

Substitute $x = 10$ into the expression for $m\angle DEG$: $m\angle DEG=(5x + 1)=(5\times10+1)=51^{\circ}$.

Step7: Find $m\angle DEF$

Substitute $x = 10$ into the expression for $m\angle DEF$: $m\angle DEF=(7x + 4)=(7\times10+4)=74^{\circ}$.

Answer:

$x = 10$
$m\angle DEG=51^{\circ}$
$m\angle DEF=74^{\circ}$