Question

given: ∠dgf=(13x - 11)°, ∠egc=(10x + 4)°. part a: find the measures of each angle below. m∠egf =, m∠ega =. part b: ∠ag choose your answer ∠egb. part c: (overline{ab}) is choose your answer (overline{fc})

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. $\angle DGF$ and $\angle EGC$ are vertical angles. So, $13x−11 = 10x + 4$.

Step2: Solve for $x$

Subtract $10x$ from both sides: $13x-10x−11=10x - 10x+4$, which gives $3x−11 = 4$. Then add 11 to both sides: $3x−11 + 11=4 + 11$, so $3x=15$. Divide both sides by 3: $x = 5$.

Step3: Find $\angle EGF$

$\angle EGF$ and $\angle DGF$ are supplementary (a straight - line is 180°). First, find $\angle DGF$: $\angle DGF=13x−11=13\times5−11=65 - 11 = 54^{\circ}$. Then $\angle EGF=180^{\circ}-\angle DGF=180 - 54=36^{\circ}$.

Step4: Find $\angle EGA$

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

Step5: Analyze perpendicularity in Part B

$\angle AGB = 180^{\circ}$ (a straight - line). Since $\angle EGB=\angle DGF = 54^{\circ}$ and $\angle AGE = 126^{\circ}$, $\angle AGE
eq90^{\circ}$, so $\angle AGE$ is not perpendicular to $\angle EGB$.

Step6: Analyze perpendicularity in Part C

Since the angle between $AB$ and $FC$ is $90^{\circ}$ (right - angle symbol in the figure), $AB$ is perpendicular to $FC$.

Answer:

Part A:
$m\angle EGF = 36^{\circ}$
$m\angle EGA = 126^{\circ}$
Part B: not perpendicular to
Part C: perpendicular to