Suggested languages for you:

Americas

Europe

Q12.

Expert-verified
Found in: Page 207

### Geometry

Book edition Student Edition
Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen
Pages 227 pages
ISBN 9780395977279

# Write proof in two-column form.Given: $\overline{QR}$, $\overline{ST}$ bisect each other. Prove: $\angle XRT>m\angle S$.

 Statement Proof 1. $QV=VR$, $SV=VT$ Given 2. $\angle QVS=\angle RVT$ Vertically opposite angles 3. $△QSV\cong △VRT$ SAS 4. $\angle T=\angle S$ CPCT 5. $\angle XRT=\angle RVT+\angle S$ 6. $m\angle XRT>m\angle S$ Property of Inequality If $a=b+c$ and $c>0$ then $a>c$.
See the step by step solution

## Step 1. Draw the given diagram and define the congruent property of triangles

Two triangles $QSV$ and $VRT$ as shown below are the given triangles.

If two sides and one angle equal of two triangles are equal then from SAS corresponding triangles will be congruent triangles.

Since QR and ST bisect each other, therefore $QV=VR$, $SV=VT$.

The vertically opposite angles are always equal.

Therefore, $\angle QVS=\angle RVT$. Then by SAS, $△QSV\cong △VRT$.

## Step 2. Property of congruent triangles

Since $△QSV\cong △VRT$ then by CPCT $\angle T=\angle S$.

Use exterior angle property as follows:

$\begin{array}{l}\angle XRT=\angle RVT+\angle T\\ \angle XRT=\angle RVT+\angle S\end{array}$

## Step 3. Prove the statement

If $a=b+c$ and $c>0$ then $a>c$.

Since $\angle XRT=\angle RVT+\angle S$ therefore $m\angle XRT>m\angle S$

Hence proved.