Question

Given:$\overline{\mathrm{PQ}}\cong \overline{\mathrm{PR}}\mathbf{;}\mathbf{}\overline{\mathrm{TR}}\cong \overline{\mathrm{TS}}$

Which one(s) of the following must be true?

(1)$\overline{\mathrm{ST}}\mathbf{\left|}\mathbf{\right|}\overline{\mathrm{QP}}$

(2)$\overline{\mathrm{ST}}\cong \overline{\mathrm{QP}}$

(3)$\angle \mathbf{T}\cong \angle \mathbf{P}$

Short answer

Only statement (1) and (3) are true.

Step-by-step solution

Step 1. Apply isosceles triangle theorem.

If the two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

It is given that,$\overline{PQ}\cong \overline{PR}$ and$\overline{TR}\cong \overline{TS}$

Therefore,$\angle Q\cong \angle PRQ$andwidth="96">∠TRS≅∠S

Step 2. Vertically opposite angles.

Angles formed due intersection of two lines are vertically opposite angles and they are equal in measures. From the figure it can be observed that $\angle PRQ$and$\angle TRS$ are vertically opposite angles. Therefore, $\angle PRQ\cong \angle TRS$.

Step 3. Transitive property of congruence.

If$\angle A\cong \angle B$ and$\angle B\cong \angle C$ then $\angle A\cong \angle C$.

Since, $\angle Q\cong PRQ$, $\angle PRQ\cong \angle TRS$and $\angle TRS\cong \angle S$implies that $\angle Q\cong \angle S$.

Step 4. Description of step.

If two lines are intersected by a transversal and alternate interior angles are congruent then the lines are parallel.

Therefore,$\overline{ST}\left|\right|\overline{QP}$and$\angle T\cong \angle P$.

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\overline{PQ}\cong \overline{PR}$	Given
$\angle Q\cong PRQ$	Isosceles triangle theorem
$\overline{TR}\cong \overline{TS}$	Given
$\angle TRS\cong \angle S$	Isosceles triangle theorem
$\angle PRQ\cong \angle TRS$	Vertically opposite angles
$\angle Q\cong \angle S$	Transitive property
$\overline{ST}\left|\right|\overline{QP}$	If two lines are intersected by a transversal and alternate interior angles are congruent then the lines are parallel.
$\angle T\cong \angle P$	Alternate interior angles

Therefore, only (1) and (3) statements are true.