Congruent Triangles Triangle ABC and Triangle DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC.If AD is extended to intersect BC at P,show that
(a)Triangle ABD ≈ Triangle ACD
(b)Triangle ABP ≈ Triangle ACP
(c)AP bisects angle a as well as angle d
(d)AP is the perpendicular bisector of BC
 A: (a) For the congruence of $\triangle ABD$ and $\triangle ACD$, use the fact that the sides match. The criterion, in North American schools, is often called SSS.
(b) It follows from the result in (a) that $\angle BAP=\angle CAP$. Then what you need should follow from the congruence criterion  often called SAS (side, angle, side). 
(c) The fact that $\angle BAP=\angle CAP$, that is, the fact that $AP$ bisects $\angle BAC$, has already been mentioned and proved. Essentially the same argument works for $\angle BDC$. 
(d) Enough has been proved above to make this straightforward. 
A: *

*Two figures are congruent, if they are of the same shape and of the same size.

*Two circles of the same radii are congruent.

*Two squares of the same sides are congruent.

*If two triangles ABC and PQR are congruent under the correspondence A – P,
B-Q and C-R, then symbolically, it is expressed as Δ ABC =
Δ PQR.

*SAS Congruence Rule: If two sides and the included angle of one triangle are 
equal to two sides and the included angle of the other triangle, then the two 
triangles are congruent. (Axiom: This result cannot be proved with the help of 
previously known results.)

*ASA Congruence Rule: If two angles and the included side of one triangle are 
equal to two angles and the included side of the other triangle, then the two 
triangles are congruent (ASA Congruence Rule).
Construction: Two triangles are given as follows, where Triangle ABC = Triangle DEF and 
Triangle ACB = Triangle DFE . Sides AB=DE
To Prove:Triangle ABC = Triangle DEF
Proof: Triangle ABC = Triangle DEF (given)
AB = DE
AC= DF (Sides opposite to corresponding angles are in the same ratio as ratio 
of angles)
Hence, by SAS congruence rule Triangle ABC = Triangle DEF is proved.

