Question

Name the definition, postulate or theorem that justifies the statement.

If $\overrightarrow{SW}$bisects ∠VST, then $\mathit{m}\mathit{\angle }\mathit{W}\mathit{S}\mathit{T}\mathit{=}\frac{1}{2}\mathit{m}\mathit{\angle }\mathit{V}\mathit{S}\mathit{T}$.

Short answer

The theorem that justifies the statement is the angle bisector theorem which states that if$\overrightarrow{BX}$ is the bisector of∠ABC, then $\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{X}\mathit{=}\frac{1}{2}\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{C}$, $\mathit{m}\mathit{\angle }\mathit{X}\mathit{B}\mathit{C}\mathit{=}\frac{1}{2}\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{C}$and $\angle ABX\cong \angle XBC$.

Step-by-step solution

Step 1. Write the angle bisector theorem.

The angle bisector theorem states that if$\overrightarrow{BX}$ is the bisector of∠ABC, then $\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{X}\mathit{=}\frac{1}{2}\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{C}$, $\mathit{m}\mathit{\angle }\mathit{X}\mathit{B}\mathit{C}\mathit{=}\frac{1}{2}\mathit{m}\mathit{\angle }\mathit{A}\mathit{B}\mathit{C}$and $\angle ABX\cong \angle XBC$.

Step 2. Description of step.

It is being given that if$\overrightarrow{SW}$bisects∠VST, therefore by using the angle bisector theorem it can be said that $m\angle WST\mathbf{=}\frac{1}{2}m\angle VST$.

Step 3. Write the theorem that justifies the statement.

The theorem that justifies the statement is the angle bisector theorem which states that if$\overrightarrow{BX}$ is the bisector of∠ABC, then $m\angle ABX=\frac{1}{2}m\angle ABC$, $m\angle XBC=\frac{1}{2}m\angle ABC$ and $\angle ABX\cong \angle XBC$.