Question

for the following intersection, the measurements of the indicated angles are represented by the expressions ∠a=(x + 46)° and ∠c=(4x + 31)°. what are the numerical angle measurements of ∠a and ∠c?

Explanation:

Step1: Set angles equal

Since $\angle a$ and $\angle c$ are vertical - angles, they are equal. So we set up the equation $x + 46=4x + 31$.
$x + 46=4x + 31$

Step2: Solve for x

Subtract $x$ from both sides: $46 = 3x+31$. Then subtract 31 from both sides: $46 - 31=3x$, so $15 = 3x$. Divide both sides by 3: $x = 5$.
$46-31 = 3x$
$15=3x$
$x = 5$

Step3: Find measure of $\angle a$

Substitute $x = 5$ into the expression for $\angle a$: $\angle a=(x + 46)^{\circ}=(5 + 46)^{\circ}=51^{\circ}$.
$\angle a=(5 + 46)^{\circ}=51^{\circ}$

Step4: Find measure of $\angle c$

Since $\angle a=\angle c$, $\angle c = 51^{\circ}$.

Answer:

$\angle a = 51^{\circ}$
$\angle c = 51^{\circ}$