Answer to Question #125942 in Geometry for zanele

Question

Suggest a sequence of statements that logically prove that the angles opposite the

congruent sides of an isosceles triangle are equal in size. Explain each statement simply

and clearly. Use illustrations throughout to enhance your explanations.

Answer 1

Isosceles Triangle Theorem states:

 If two sides of a triangle are congruent, then angles opposite those sides are congruent.

THEOREM PROOF :

Consider an isosceles triangle PQR where PR=QR . We need to prove that the angles opposite to the sides PR and QR are equal , that is ∠RPQ = ∠RQP .

We first draw a bisector of ∠PRQ and named it RS.

Now in ∆PRS and ∆QRS , we have

PR = QR (given)

∠PRS = ∠QRS (by construction)

CD = CD (common to both)

Thus, ∆PRS ≅ ∆QRS (by SAS congruency)

So, ∠RPQ = ∠RQP (by CPCTC)

where SAS= (side-angle-side)

and CPCTC = (Corresponding parts of congruent triangles are congruent)

HENCE PROVED .