# 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.

Solution:

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

Proof:

 Statement1. 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 = ∠PRS6. ∆PQS ≅ ∆PRS7. ∠QPS = ∠RPS = z° (Suppose).8. ∠QST = ∠PQS + ∠QPS = x° - y° + z°9. Similarly, ∠RST = ∠PRS + ∠RPS = x° - y° + z° 10. ∠QST = ∠RST (Proved) Reason1. PQ = PR2. SQ = SR3. Subtracting statement 2 from statement 1.4. From statement 3.5. (i) Given. (ii) Given.(iii) From statement 4.6. By SAS criterion7. CPCTC8. 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.