Question

given
∠dgf=(13x - 11)°
∠egc=(10x + 4)°
part a: find the measures of each angle below.
m∠egf=
m∠ega=
part b: ∠agd
∠egb

Explanation:

Step1: Use vertical - angle property

Since $\angle DGF$ and $\angle EGC$ are vertical angles, they are congruent. So we set up the equation $13x - 11=10x + 4$.
$13x-10x=4 + 11$
$3x=15$
$x = 5$.

Step2: Find $\angle EGF$

We know that $\angle EGF$ and $\angle EGC$ are complementary (because $\angle BGC = 90^{\circ}$). First, find $\angle EGC$ by substituting $x = 5$ into the expression for $\angle EGC$. $\angle EGC=10x + 4=10\times5+4=54^{\circ}$. Then $\angle EGF = 90^{\circ}-\angle EGC=90 - 54=36^{\circ}$.

Step3: Find $\angle EGA$

$\angle EGA$ and $\angle EGB$ are supplementary. $\angle EGB=\angle EGC = 54^{\circ}$ (vertical angles), so $\angle EGA=180 - 54=126^{\circ}$.

Step4: Compare $\angle AGD$ and $\angle EGB$

$\angle AGD$ and $\angle EGB$ are vertical angles, so they are congruent.

Answer:

$m\angle EGF = 36^{\circ}$
$m\angle EGA = 126^{\circ}$
$\angle AGD$ is congruent to $\angle EGB$