Question

HELP ASAP

Given: RT || SP, RQ ≅ QP, RP bisects ST at Q
Prove: ΔRQT ≅ ΔPQS

Tamir is working to prove the triangles congruent using SAS. After stating the given information, he states that TQ ≅ QS by the definition of segment bisector. Now he wants to state that ∠RQT ≅ ∠PQS. Which reason should he use?

Answer 1

∠RQT ≅ ∠PQS (by the vertical angles theorem)

Explanation:

Given: RT || SP, RQ ≅ QP, RP bisects ST at Q

To prove: ΔRQT ≅ ΔPQS

Proof:

It is given that RT || SP, RQ ≅ QP, RP bisects ST at Q, thus TQ=QS

From ΔRQT and ΔPQS, we have

RQ ≅ QP (Given)

∠RQT ≅ ∠PQS (by the vertical angles theorem)

TQ ≅ QS (by the definition of segment bisector)

Therefore, by SAS rule of congruency, we have

ΔRQT ≅ ΔPQS

Hence proved.

Answer 2

SAS theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

1. RQ ≅ QP (given)

2. If RP bisects ST at Q, then TQ ≅ QS (by the definition of segment bisector).

3. ∠RQT, ∠PQS are vertical angles (angles that are vertically opposite to each other when lines PR and ST intersect, Q is the common vertex).

∠RQT ≅ ∠PQS (by the vertical angles theorem).

Vertical angles theorem states that vertical angles are always congruent.

According to SAS theorem, ΔRQT ≅ ΔPQS.

Answer: vertical angles theorem