Chapter 6.1, Problem 12WE

To determine

To Prove: $m\angle XRT>m\angle S$ in two-column form shown in figure such that $\overline{QR}\text{ and }\overline{ST}$ bisect each other.

Answer to Problem 12WE

Explanation of Solution

Given:

Figure is shown as.

  

  $\overline{QR}\text{ and }\overline{ST}$ bisect each other.

Calculation:

Since, $\angle$ is exterior angle of the triangle, therefore.

  $\begin{array}{cc}\text{Statements}& \text{Reasons}\\ \begin{array}{l}QV=VR\\ SV=VT\end{array}& \text{Given}\\ \angle QVS=\angle RVT& \text{Vertically Opposite angles}\\ \Delta QVS\cong \Delta RVT& \text{By Side angle-Side}\left(\text{SAS}\right)\text{ Postulate}\\ \begin{array}{l}\angle T=\angle S\\ \angle XRT=\angle RVT+T\\ \angle XRT=\angle S+\angle RVT\\ m\angle XRT>m\angle S\end{array}& \begin{array}{l}\text{By Congruent parts of coongruent triangles}\left(\text{CPCT}\right)\\ \text{Exterior angle}\\ \angle T=\angle S\\ \text{If a=b+c and c>0, then a>c}\end{array}\end{array}$

Therefore, it has been proved.