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:
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. 
From Problem on Two Isosceles Triangles on the Same Base to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.