Question

In triangles \[\vartriangle ABC,\vartriangle DEF\], \[AB = FD\] and \[\angle A = \angle D\]. The two triangles will be congruent by SAS axiom if
A. \[BC = EF\]
B.\[AC = DE\]
C.\[AC = EF\]
D.\[BC = DE\]

Answer 1

Hint: Here we use the definition of SAS congruence rule which states that when two sides and the included angle between the sides of one triangle are equivalent to corresponding sides and the including angle between the corresponding sides then two triangles are said to be congruent.
* Included angle means the angle between the sides.

Complete step-by-step answer:
We first draw figures with two triangles namely \[\vartriangle ABC,\vartriangle DEF\] which have one pair of equal sides \[AB = FD\] and one pair of equal angles \[\angle A = \angle D\].
     

We know SAS congruence rule gives two triangles congruent if we have two sides of a triangle equal to two sides of another triangle and the angle between two equal sides of one triangle is equal to angle between two equal sides of the other triangle.
We have the set of angles which are equal \[\angle A = \angle D\].
So, we look for the sides of \[\vartriangle ABC\] which have \[\angle A\] in between them and the sides of \[\vartriangle DEF\] which have \[\angle D\] in between them.
Sides having \[\angle A\] between them are \[AB,AC\]
Sides having \[\angle D\] between them are \[DE,FD\]
We get four sides \[AB,AC\] and \[DE,FD\].
We know from the question statement that \[AB = FD\]
So, remaining sides from four sides are \[AC,DE\]
For triangles to be congruent by SAS congruence rule, these two sides should be equivalent to each other.
Thus, \[AC = DE\]
So, the correct answer is “Option B”.

Note: Many students make the mistake of assuming SAS meaning any two sides and one angle but they should keep in mind the proper pattern that two sides and the included angle between them have to be equal. It is useful to draw diagrams for these types of questions.