Question

In triangles $\Delta ABC$ and $\Delta DEF$, $\angle B = \angle E$, $\angle F = \angle C$ and $AB = 3DE$. Then, the two triangles are:
A. Congruent but not similar.
B. Similar but not congruent.
C. Neither congruent or similar.
D. Congruent as well as similar.

Answer 1

Hint: Similarity of Triangles: If two corresponding angles of a pair of triangles are equal, then the triangles are similar.
Congruence of Triangles: If two triangles are similar and a pair of corresponding sides are also equal, then the triangles are congruent (identical).
Compare the sides of both the triangles, opposite to the corresponding / equal angles, with each other and see if they are equal or proportional.

Complete answer:
Since one pair of angles in triangle $\Delta ABC$ is equal to a pair of angles of the triangle $\Delta DEF$, both the triangles are definitely similar (AA rule of similarity says that If two corresponding angles of a pair of triangles are equal, then the triangles are similar).

 \[\angle B = \angle E\] (given)
 \[\angle C = \angle F\] (given)
$AB = 3DE$
But the triangles are not congruent because the sides AB and DE, which are opposite to the equal corresponding angles $\angle C$ and $\angle F$, are not equal.
The correct answer is, therefore, B. Similar but not congruent.
So, the correct answer is “Option B”.

Note: Congruent triangles are also similar to each other. A pair of triangles which are congruent but not similar, is not possible.
Other Rules for Similarity of two triangles:
SAS: If two corresponding sides of a pair of triangles are proportional (have the same ratio) and the angle between them is also equal, then the triangles are similar.
SSS: If all the three pairs of corresponding sides of two triangles are proportional (have the same ratio), then the triangles are similar.