Question

angle def is bisected by ray eg. angle feg = 2x + 5, and angle deg = 3x - 7. find the measure of angle def.

Explanation:

Step1: Set equal the two angle - measures

Since ray EG bisects angle DEF, then $\angle FEG=\angle DEG$. So we set up the equation $2x + 5=3x - 7$.

Step2: Solve for x

Subtract $2x$ from both sides: $2x+5-2x=3x - 7-2x$, which gives $5=x - 7$. Then add 7 to both sides: $5 + 7=x-7 + 7$, so $x = 12$.

Step3: Find the measure of one of the bisected angles

Substitute $x = 12$ into the expression for $\angle FEG$ (we could also use $\angle DEG$). $\angle FEG=2x + 5=2\times12+5=24 + 5=29$.

Step4: Find the measure of angle DEF

Since $\angle DEF=\angle FEG+\angle DEG$ and $\angle FEG=\angle DEG$, then $\angle DEF = 2\angle FEG$. So $\angle DEF=2\times29 = 58$.

Answer:

$58$