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Given: ST;ST||QP

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

(1)PQ

(2)PRQR

(3) R is the midpoint of PT

Short Answer

Expert verified

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

Step by step solution

01

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 thatTR is opposite toS andRS is opposite toT.

As ST, TRRS.

02

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,ST||QP andTP;QS.

03

Step 3. Transitive property of congruence.

IfAB andBC then AC.

Since, PT, TSand SQimplies that PQ.

04

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 QRis opposite to Pand PRis opposite to Q.

As PQ, PRQR.

05

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement

Reason

ST

Given

TRRS

Converse of isosceles triangle theorem

ST||QP

Given

TP;QS

Alternate interior angles

PQ

Transitive property

PRQR

Converse of isosceles triangle theorem

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

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