Question

what term refers to the pair of labeled angles? are these angles congruent? if so, find the value of x. if not, explain why not.

Explanation:

Step1: Identify angle - type

The angles $(10x - 19)^{\circ}$ and $(3x+ 16)^{\circ}$ are vertical angles. Vertical angles are congruent.

Step2: Set up the equation

Since vertical angles are equal, we set up the equation $10x-19 = 3x + 16$.

Step3: Isolate the variable terms

Subtract $3x$ from both sides: $10x-3x-19=3x - 3x+16$, which simplifies to $7x-19 = 16$.

Step4: Isolate the variable

Add 19 to both sides: $7x-19 + 19=16 + 19$, so $7x=35$.

Step5: Solve for x

Divide both sides by 7: $x=\frac{35}{7}=5$.

Answer:

The term for the pair of labeled angles is vertical angles. They are congruent and $x = 5$.