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.
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.
(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.