Question

Given: $\angle \mathbf{S}\cong \angle \mathbf{T}$;$\overline{\mathrm{ST}}\mathbf{\left|}\mathbf{\right|}\overline{\mathrm{QP}}$

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

(1)$\angle \mathbf{P}\cong \angle \mathbf{Q}$

(2)$\overline{\mathrm{PR}}\cong \overline{\mathrm{QR}}$

(3) R is the midpoint of $\overline{\mathrm{PT}}$

Short answer

Only statement (1) and (2) are true.

Step-by-step solution

Step 1. Converse of isosceles theorem.

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

From the given figure it can be observed that$\overline{TR}$ is opposite to$\angle S$ and$\overline{RS}$ is opposite to$\angle T$.

As $\angle S\cong \angle T$, $\overline{TR}\cong \overline{RS}$.

Step 2. Description of step.

If two parallel lines are intersected by a transversal then the angles made by transversal with respect to lines are congruent and are called alternate angles.

As,$\overline{ST}\left|\right|\overline{QP}$ and$\angle T\cong \angle P;\angle Q\cong \angle S$.

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, $\angle P\cong \angle T$, $\angle T\cong \angle S$and $\angle S\cong \angle Q$implies that $\angle P\cong \angle Q$.

Step 4. Converse of isosceles theorem.

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

From the given figure it can be observed that $\overline{QR}$is opposite to $\angle P$and $\overline{PR}$is opposite to $\angle Q$.

As $\angle P\cong \angle Q$, $\overline{PR}\cong \overline{QR}$.

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\angle S\cong \angle T$	Given
$\overline{TR}\cong \overline{RS}$	Converse of isosceles triangle theorem
$\overline{ST}\left|\right|\overline{QP}$	Given
$\angle T\cong \angle P;\angle Q\cong \angle S$	Alternate interior angles
$\angle P\cong \angle Q$	Transitive property
$\overline{PR}\cong \overline{QR}$	Converse of isosceles triangle theorem

Therefore, only (1) and (2) statements are true.