Question

Which of the following statements is false?
A) All isosceles triangles are similar.
B) All equilateral triangles are similar.
C) All circles are similar.
D) None of the above.

Answer 1

Hint: We will first understand the criteria for similarity and what does similarity represent. After that, we will go in with all the options one by one to understand whether they follow the criteria or not and thus have the required answer.

Complete step-by-step answer:
We know that “Two figures are said to be similar if they have the same shape”. In other words, if we have two figures with us and we can achieve one from another by just resizing the figure keeping the ratio of sides still constant, we will say that the figures are similar. In triangles, we also have some criteria for similarity of triangles that is AA and SAS.
AA similarity criterion: If two of the corresponding angles of the triangles are equal, then the triangles are said to be similar.
SAS similarity criterion: If two of the corresponding sides and one corresponding angle is equal, then the triangles are said to be similar.
Now, let us go into the options one by one.
Option A: All isosceles triangles are similar
Let us assume that it is correct, then it must be true for all the isosceles triangles.

If these isosceles triangles are similar, then they must follow either one of the AA or SAS criterion but they cannot because none of the angles is equal to one another. Hence, our assumption was incorrect.
Hence, option (A) is false.
Option B: All equilateral triangles are similar
We see that, we have a criterion that is AA which must be satisfied by all of the equilateral triangles as they all have ${60^ \circ }$ angles. Hence, the statement is correct.
Hence, option (B) is true.
Option C: All circles are similar
We have discussed in the start that two figures are similar if they both have the same shape. We know that, in a circle only the radius can change, otherwise everything remains the same because it has no sides and no angles. Therefore, we can always resize one circle to get another and hence, the circles are similar.
Hence, option (C) is true.

Hence, option (A) is the required answer.

Note: The students must know that AA criterion in triangles is the same as AAA criterion, because if two of the angles are equal in triangles, the third must also be equal to corresponding angles because of the angle sum property. Therefore, to make it easier for us, we write AA instead of AAA.
The students must know that there is a lot of difference between similarity and congruence. If two figures are congruent, we can overlap them with one another without the other one showing itself from the bottom whereas in similarity, we can resize them without changing the side ratio to get them overlapping.
“All congruent figures are similar but converse may not be true.”