# Theorem on Isosceles Triangle

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:

 Statement1. In ∆XQY and ∆XPZ,(i) XY = XZ.(ii) XQ = XP.(iii) ∠QXY = ∠PXZ. 2. ∆XQY ≅ ∆XPZ. 3. QY = PZ. (Proved) Reason1. (i) Given.(ii) Given.(iii) Vertically opposite angles. 2. By SAS criterion. 3. CPCTC.