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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:
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Statement 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) |
Reason 1. PQ = PR 2. SQ = SR 3. Subtracting statement 2 from statement 1. 4. From statement 3. 5. (i) Given. (ii) Given. (iii) From statement 4. 6. By SAS criterion 7. CPCTC 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. |
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