Question

Prove that two triangles are congruent if any two sides and the
included angle of one triangle is equal to any two sides and the included angle of other triangle.

Answer 1

You are asked to prove SAS theorem proof ,I will give you the proof
But I don't think you will understand these are theorem's proof ,that are beyond your scope
It is betrer you just learn that it is a theorem which states if any two sides and the
included angle of one triangle is equal to any two sides and the included angle of other triangle,
then the triangles are congruent


The Proof For the proof of the side angle side theorem (SAS) we need: (1) The Axiom of Movement (2) The Mid-Point theorem 1. Axiom of Movement Any geometric figure may be moved from one place to another without changing its size or shape. 2. Mid-Point Theorem The straight line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of it. Theorem Two triangles are congruent if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other. Given. ABC, DEF are triangles such that. AB = DE AC = DF < BAC = < EDF To prove. ∆s ABC, DEF are congruent Construction: By the axiom permitting motion, apply ∆DEF to ∆ABC so that D falls on A and E falls on B, and EF along BC. Rotate ∆DEF about A by 180o to form the cross section of the double cone in fig.(ii). Join EC and, from the coincident point of A and D of the cross section, draw a line HK to bisect EC at K, fig (iii). We write H = (A;D) so that H refers to A, or to D if we refer to ∆ABC, or ∆DEF. The theorem will be proved if we can show that BC = EF. Proof: < BHC = < EHF (given) But these are vertically opposite angles  Lines BE, CF are straight lines In ∆BCE, BH = EH (given) CK = EK (construction)  HK bisects BE and EC.  In particular, HK = ½ BC (Mid-Point Theorem) Similarly in ∆EFC, HK bisects CF and CE (Construction)  HK = ½ EF (Mid-Point Theorem)  BC = EF (Things equal to the same thing are equal to each other)  ∆s ABC, DEF are congruent