Here we will prove that the equal sides YX and ZX of an isosceles triangle XYZ are produced beyond the vertex X to the points P and Q such that XP is equal to XQ. QY and PZ are joined. Show that QY is equal to PZ.
Solution:
In ∆XYZ, XY = XZ. YX and XZ are produced to P and Q respectively such that XP = XQ. Q, Y and P, Z are joined.
To prove: QY = PZ.
Proof:
Statement 1. In ∆XQY and ∆XPZ, (i) XY = XZ. (ii) XQ = XP. (iii) ∠QXY = ∠PXZ.
2. ∆XQY ≅ ∆XPZ. 3. QY = PZ. (Proved) 
Reason 1. (i) Given. (ii) Given. (iii) Vertically opposite angles.
2. By SAS criterion. 3. CPCTC. 
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