Question

the angle measurements in the diagram are represented by the following expressions.
∠a = 8x + 78°
∠b = 2x + 114°

Explanation:

Step1: Identify angle - relationship

Since $\angle A$ and $\angle B$ are vertical angles, they are equal. So we set up the equation $8x + 78=2x + 114$.

Step2: Solve for $x$

Subtract $2x$ from both sides: $8x-2x + 78=2x-2x + 114$, which simplifies to $6x+78 = 114$. Then subtract 78 from both sides: $6x+78 - 78=114 - 78$, getting $6x=36$. Divide both sides by 6: $x=\frac{36}{6}=6$.

Step3: Find the measure of $\angle A$ or $\angle B$

Substitute $x = 6$ into the expression for $\angle A$ (we could also use $\angle B$). $\angle A=8x + 78=8\times6+78=48 + 78=126^{\circ}$.

Answer:

$x = 6$, $\angle A=\angle B = 126^{\circ}$