Question

If two triangles are congruent, are they similar? Please explain why or why not.

Answer 1

Hint: Two triangles are congruent when their corresponding sides are equal and all the corresponding angles equal in measurements. It means triangles are congruent when they superimpose each other. Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional.

Complete step by step solution:
Let there are two triangles \[\vartriangle ABC\] and \[\vartriangle DEF\] that are congruent to each other. Then the two triangles look exactly similar to each other and also superimpose each other because the length of their corresponding sides will be equal and their corresponding angles will be equal. So, they will be the exact photocopy of each other.
Two triangles are similar when their corresponding sides are proportional and their corresponding angles are equal. It is not necessary that the lengths of their corresponding sides should be equal. Since their corresponding angles are equal and corresponding sides are proportional they will look similar to each other in shape but their size may not be the same.
So, congruent means same shape and same size and similar means same shape, sizes can be different or same.
This implies that if the triangles are congruent, they will also be similar but if they are similar then it is not necessary that they will be congruent. For example: -

These two triangles, \[\vartriangle ABC\] and \[\vartriangle DEF\] are congruent to each other by SAS property. We can see that they are also looking similar. Also suppose two triangles are similar by AA property. We can see that they are of the same shape but not of the same size. So, they are not congruent.

Note:
One of the most important things to note is that while writing which triangle is congruent to which triangle, we should write the corresponding sides and angles in the same order. For example, in the first figure \[\vartriangle ABC\] and \[\vartriangle DEF\] are congruent and not the triangles \[\vartriangle BAC{\text{ and }}\vartriangle {\text{DEF}}\]. The same rule applies for the similarity of triangles also.