Question

Fill in the blanks with the correct option. All _______ triangles are similar.
(a) Right angled
(b) Isosceles
(c) Equilateral
(d) Reflex

Answer 1

Hint: Study the definitions and properties of all the triangles and all the similarity criterions by which any two triangles are proved to be similar. Check for the general case of all the triangles given the options. Use the information that, A – A – A similarity criteria is one of the most important criteria to prove that two triangles are similar.

Complete step by step answer:
Let us come to the question. We have been provided with the options: - Right angled, Equilateral and Reflex triangles. Let us see their property one – by – one.
(i) Right angled triangle: -
It is a type of triangle that contains one angle equal to \[{{90}^{\circ }}\] and two acute angles. Two right angle triangles can be similar by various criterias. Here, let us talk about A – A – A criteria which means all angles of one triangle should be the same corresponding to the angles of the second triangle. Let us take an example: -

Let us consider two right angle triangles ABC and DEF. Here, \[\angle B=\angle E={{90}^{\circ }}\]. But angles A and C do not match with the corresponding angles D and F. Hence in general we can say that two right angle triangles are not always similar.
(ii) Isosceles triangle: -
It is a type of triangle in which two sides or two angles are equal. Let us check if they are always similar or not.

We consider two isosceles triangles ABC and DEF. Here, we can see that they are not similar because none of the angles are matching. Hence, in general isosceles triangles are not similar always.
(iii) Equilateral triangles: -
It is a type of triangle in which all the sides are equal and each angle is \[{{60}^{\circ }}\]. Let us check for their similarity.

We consider two triangles ABC and DEF which are equilateral. Here, we can see that all the angles of \[\Delta ABC\] and \[\Delta DEF\] are \[{{60}^{\circ }}\] and this will be true for any two equilateral triangles. So, in general all equilateral triangles are always similar by A – A – A criteria.
(iv) Reflex triangle: -
It is a type of triangle in which one angle is obtuse and the other two angles are acute. Let us check for their similarity.

We consider two obtuse angle triangles ABC and DEF. Here, we can see that only one of their angles are matching and the rest does not match. Hence, in general reflex triangles are not always similar.
Therefore, finally we can conclude that all equilateral angle triangles are always similar.

So, the correct answer is “Option C”.

Note: You may note that to eliminate the other three options. We need only one example to prove the statement wrong. So, we took such an example which can prove that only equilateral triangles can be similar all the time. Always remember that you do not have to try various types of examples or situations, just think of a situation which can prove the statement false. Remember that, in mathematics even one small example can prove a statement wrong.