Question

if two sides and median bisecting one of these side of a triangle are equal respectively to the two sides and the median bisecting one of these sides of another triangle, prove that two triangles are congruent

Answer 1

AX is median. so it bisects BC
i.e. BX = 1/2 BC. -------------------1

similarly, DY is median. so it bisects EF
i.e EY = 1/2 EF
but we know, BC = EF
EY = 1/2 BC. ------------------2

from equations 1 and 2 by transitive property,
BX = EY

in triangles ABX and DEY,
by SSS congrueny criterion,
they are congruent.

so, angle A = angle E ( corresponding parts of congruent triangles )

so now, consider triangles ABC and DEF
by SAS congruence criterion, ( since AB = DE ; BC = EF ; angle A = angle E)
they are congruent


hope it helps