Problem on Two Isosceles Triangles on the Same Base

Here we will prove that ∆PQR and ∆SQR are two isosceles triangles drawn on the same base QR and on the same side of it. If P and S be joined, prove that each of the angles ∠QPR and ∠QSR will be divided by the line PS into two equal parts.

Two Isosceles Triangles on the Same Base


Given: PQ = PR and SQ = SR to prove ∠QPS = ∠ RPS and ∠QST = ∠RST



1. In ∆PQR, ∠PQR = ∠PRQ = x° (Suppose)

2. In ∆SQR, ∠SQR = ∠SRQ = y° (Suppose).

3. ∠PQR - ∠SQR = ∠PRQ - ∠SRQ = x° - y°

4. Therefore, ∠PQS = ∠PRS = x° - y°

5. In ∆PQS and ∆PRS,

(i) PQ = PR

(ii) SQ = SR

(iii) ∠PQS = ∠PRS

6. ∆PQS ≅ ∆PRS

7. ∠QPS = ∠RPS = z° (Suppose).

8. ∠QST = ∠PQS + ∠QPS = x° - y° + z°

9. Similarly, ∠RST = ∠PRS + ∠RPS = x° - y° + z°

10. ∠QST = ∠RST (Proved)


1. PQ = PR

2. SQ = SR

3. Subtracting statement 2 from statement 1.

4. From statement 3.


(i) Given.

(ii) Given.

(iii) From statement 4.

6. By SAS criterion


8. The exterior angle of a triangle is equal to the sum of the interior opposite angles.

9. As above.

10. From statements 8 and 9.

9th Grade Math

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