Chapter 4.3, Problem 1WE

To determine

To prove: O is the midpoint of $\overline{RQ}$ .

Explanation of Solution

Given information: $\angle P\cong \angle S$ and O is the midpoint of $\overline{PS}$ .

Proof:

  

Statements	Reasons
$\angle P\cong \angle S$	Given
O is the midpoint of $\overline{PS}$	Given
$\overline{PO}\cong \overline{SO}$	As O is the midpoint of $\overline{PS}$ and midpoint is a point on a line segment that divides it into two congruent line segments
$\angle POQ\cong \angle SOR$	Vertically opposite angles
$\Delta POQ\cong \Delta SOR$	ASA congruence criteria (according to ASA congruence criteria, two triangles are said to be congruent if two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle)
$\overline{QO}\cong \overline{RO}$	Corresponding parts of congruent triangles are equal
O is the midpoint of $\overline{RQ}$	As $\overline{QO}\cong \overline{RO}$ and midpoint is a point on a line segment that divides it into two congruent line segments