Question

1. assume that ∠amz and ∠cmx are straight angles. use the definition of linear pairs to show that ∠amc ≅ ∠xmz in the diagram. (you are proving the vertical - angle theorem, so you can’t use it. also, you do not need numbers to show this.)

Explanation:

Step1: Define linear - pair property

Since $\angle AMZ$ is a straight - angle, $\angle AMC+\angle CMZ = 180^{\circ}$ (by the definition of a linear pair, where two adjacent angles that form a straight line add up to $180^{\circ}$).

Step2: Define linear - pair property for another pair

Since $\angle CMX$ is a straight - angle, $\angle XMZ+\angle CMZ = 180^{\circ}$ (by the definition of a linear pair).

Step3: Use the substitution property

From $\angle AMC+\angle CMZ = 180^{\circ}$ and $\angle XMZ+\angle CMZ = 180^{\circ}$, we can say that $\angle AMC+\angle CMZ=\angle XMZ+\angle CMZ$.

Step4: Subtract $\angle CMZ$ from both sides

Subtracting $\angle CMZ$ from both sides of the equation $\angle AMC+\angle CMZ=\angle XMZ+\angle CMZ$, we get $\angle AMC=\angle XMZ$, which means $\angle AMC\cong\angle XMZ$.

Answer:

$\angle AMC\cong\angle XMZ$