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:**

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

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