Question

Which condition for congruence is used to prove that the following pairs of triangles are congruent?

A. ASA
B. AAS
C. SAS
D. SSS

Answer 1

Hint: Prove them to be congruent either by proving their three sides are equal or three angles are equal or if they are fulfilling the any three conditions of sides or angles.
Two figures are congruent if they have exactly the same size and shape. There are four criteria to test whether the given triangles are congruent or not: Side-Side-Side (SSS), Angle-Angle-Angle (AAA), Side-Angle-Side (SAS), Angle-Side-Angle (ASA).
In this question since only the sides of the triangles are given so the two triangles can be congruent only by their sides so we will check whether the given value of sides are equal for two triangles or not.

Complete step by step answer:
Given the length of the sides in the \[\Delta ABC\]
\[
  AB = 8cm \\
  BC = 12cm \\
  CA = 9cm \\
 \]
The length of the sides in the \[\Delta DEF\]
\[
  DE = 8cm \\
  EF = 12cm \\
  FD = 9cm \\
 \]
Now we know that the two triangles can be congruent if they fulfill the condition of congruency of: Side-Side-Side (SSS), Angle-Angle-Angle (AAA), Side-Angle-Side (SAS) and Angle-Side-Angle (ASA).
From the figure we can say that the length of the sides
\[
  AB = DE = 8cm \\
  BC = EF = 12cm \\
  CA = FD = 9cm \\
 \]
Now we can say that the two triangles are congruent since they are fulfilling the criteria of Side-Side-Side (SSS) as the length of the sides of the two triangles are equal.
Hence we can say that the two triangles are congruent with Side-Side-Side (SSS) criteria.

Option D is correct.

Note: To prove two triangles to be congruent it is not necessary that equal sides or the equal angles have the same position if we flip any of the two figures then they will have same size and shape if they are congruent. In this question if we rotate the second triangle by \[{180^ \circ }\] in clockwise direction then two triangles will have the same positions.