Answer

Verified

449.1k+ views

Hint: The question is related to the similarity of triangles. Try to recall the AAA criterion for similarity of triangles and use the property that all the internal angles of an equilateral triangle measure ${{60}^{o}}$.

Complete step-by-step answer:

Before proceeding with the solution, we must understand the concept of similarity of triangles. Two triangles are said to be similar if their angles are equal and their corresponding sides are proportional. Consider the two triangles given in the figure:

The two triangles $\Delta ABC$ and $\Delta EDF$ are similar if $\dfrac{a}{e}=\dfrac{b}{d}=\dfrac{c}{f}$ . Also, if the two triangles are similar then, $\measuredangle A=\measuredangle E;\measuredangle B=\measuredangle D$ and $\measuredangle C=\measuredangle F$.

Now, we know that in an isosceles triangle, the base angles are equal. But it is not compulsory that the angles will be equal between two triangles.

In case of an equilateral triangle, the three angles of the triangle are equal and their measure is fixed, i.e. ${{60}^{o}}$ . So, in all equilateral triangles, the measure of the angles is fixed and hence, the angles of two equilateral triangles will be equal. So, two equilateral triangles will always be similar.

Hence, all equilateral triangles are similar.

Note: Congruence and similarity of triangles are not the same. A congruent triangle is necessarily similar, but a similar triangle may or may not be congruent. For congruence of triangles, the corresponding sides of the two triangles should be equal, whereas for similarity the corresponding sides of the two triangles should be proportional. Students generally get confused between the two concepts. Such confusion should be avoided.

Complete step-by-step answer:

Before proceeding with the solution, we must understand the concept of similarity of triangles. Two triangles are said to be similar if their angles are equal and their corresponding sides are proportional. Consider the two triangles given in the figure:

The two triangles $\Delta ABC$ and $\Delta EDF$ are similar if $\dfrac{a}{e}=\dfrac{b}{d}=\dfrac{c}{f}$ . Also, if the two triangles are similar then, $\measuredangle A=\measuredangle E;\measuredangle B=\measuredangle D$ and $\measuredangle C=\measuredangle F$.

Now, we know that in an isosceles triangle, the base angles are equal. But it is not compulsory that the angles will be equal between two triangles.

In case of an equilateral triangle, the three angles of the triangle are equal and their measure is fixed, i.e. ${{60}^{o}}$ . So, in all equilateral triangles, the measure of the angles is fixed and hence, the angles of two equilateral triangles will be equal. So, two equilateral triangles will always be similar.

Hence, all equilateral triangles are similar.

Note: Congruence and similarity of triangles are not the same. A congruent triangle is necessarily similar, but a similar triangle may or may not be congruent. For congruence of triangles, the corresponding sides of the two triangles should be equal, whereas for similarity the corresponding sides of the two triangles should be proportional. Students generally get confused between the two concepts. Such confusion should be avoided.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths