Question

a. In each exercise use the information given to conclude that two angles are congruent.

b. Name or state the definition or theorem that justifies your conclusion.

$\overleftrightarrow{XZ}$bisects $\angle WXY$.

Short answer

Given that$\overleftrightarrow{\mathbf{X}\mathbf{Z}}$bisects$\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Y}$.

Therefore, by using the angle bisector theorem it is obtained that:

$\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Z}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Y}$ and$\mathbf{\angle }\mathit{Z}\mathit{X}\mathit{Y}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Y}$.

Therefore, by using the above relation it can be obtained that$\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Z}\mathbf{=}\mathbf{\angle }\mathit{Z}\mathit{X}\mathit{Y}$.

As$\mathbf{\angle }\mathit{W}\mathit{X}\mathit{Z}\mathbf{=}\mathbf{\angle }\mathbf{6}$and$\mathbf{\angle }\mathit{Z}\mathit{X}\mathit{Y}\mathbf{=}\mathbf{\angle }\mathbf{7}$, therefore it can be deduced that$\mathbf{\angle }\mathbf{6}\mathbf{=}\mathbf{\angle }\mathbf{7}$.

As the angles$\mathbf{\angle }\mathbf{6}$and$\mathbf{\angle }\mathbf{7}$have equal measure therefore the angles$\angle 6$and$\angle 7$are congruent angles.

That implies,$\mathbf{\angle }\mathbf{6}\mathbf{\cong }\mathbf{\angle }\mathbf{7}$.

b. The name of the theorem that justifies the conclusion is the angle bisector theorem.

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Write the angle bisector theorem.

The angle bisector theorem states that if$\overleftrightarrow{BX}$ is the bisector of $\angle ABC$, then$\angle ABX=\frac{1}{2}\angle ABC$ and $\angle XBC=\frac{1}{2}\angle ABC$.

Step 3. Use the given information to conclude that the two angles are congruent.

Given that $\overleftrightarrow{XZ}$bisects$\angle WXY$.

Therefore, by using the angle bisector theorem it is obtained that:

$\angle WXZ=\frac{1}{2}\angle WXY$and $\angle ZXY=\frac{1}{2}\angle WXY$.

Therefore, by using the above relation it can be obtained that $\angle WXZ=\angle ZXY$.

As$\angle WXZ=\angle 6$ and $\angle ZXY=\angle 7$, therefore it can be deduced that $\angle 6=\angle 7$.

As the angles∠6 and∠7have equal measure therefore the angles∠6 and∠7 are congruent angles.

That implies, $\angle 6\cong \angle 7$.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Write the angle bisector theorem.

The angle bisector theorem states that if$\overleftrightarrow{BX}$ is the bisector of $\angle ABC$, then$\angle ABX=\frac{1}{2}\angle ABC$ and $\angle XBC=\frac{1}{2}\angle ABC$.

Part b. Step 3. Write the name of the theorem that justifies the conclusion.

The name of theorem that justifies the conclusion is angle bisector theorem.