Can you give an alternative proof of the Converse of isosceles triangl

1.	Can you give an alternative proof of the Converse of isosceles triangle theorem by drawing a line through point R and parallel to seg PQ in the above figure?
Answer» Yes. Construction: Draw line RM parallel to seg PQ through a point R. Proof: seg PQ \|\| line RM and seg PR is their transversal. [Construction] ∴ ∠PRM = ∠QPR ……..(i) [Alternate angles] seg PQ \|\| line RM and seg QR is their transversal. [Construction] ∴ ∠SRM = ∠PQR ……..(ii) [Corresponding angles] ∴ ∠PRM + ∠SRM = ∠QPR + ∠PQR [Adding (i) and (ii)] ∴ ∠PRS = ∠PQR + ∠QPR [Angle addition property]

Can you give an alternative proof of the Converse of isosceles triangle theorem by drawing a line through point R and parallel to seg PQ in the above figure?

Answer»

Yes.

Construction:

Draw line RM parallel to seg PQ through a point R.

Proof:

seg PQ || line RM and seg PR is their transversal. [Construction]

∴ ∠PRM = ∠QPR ……..(i) [Alternate angles]

seg PQ || line RM and seg QR is their transversal. [Construction]

∴ ∠SRM = ∠PQR ……..(ii) [Corresponding angles]

∴ ∠PRM + ∠SRM = ∠QPR + ∠PQR [Adding (i) and (ii)]

∴ ∠PRS = ∠PQR + ∠QPR [Angle addition property]