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Ex.7.2 Q4 Congruence of Triangles - NCERT Maths Class 7

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Question

In \(ΔABC\), \(∠A = 30^\circ\), \( ∠B = 40^\circ\) and \(∠C = 110^\circ\). In \(ΔPQR\), \(∠P = 30^\circ\), \(∠Q = 40^\circ\) and \(∠R =110^\circ\). A student says that \(ΔABC\) \(≅\) \(ΔPQR\) by \(AAA\) congruence criterion. Is he justified? Why or why not?

 Video Solution
Congruence Of Triangles
Ex 7.2 | Question 4

Text Solution

What is known?

In \(ΔABC\), \(∠A = 30^\circ\), \( ∠B = 40^\circ\) and \(∠C = 110^\circ\)

In \(ΔPQR\), \(∠P = 30^\circ\), \(∠Q = 40^\circ\) and \(∠R =110^\circ\)

What is the unknown?

Justification that \(ΔABC\) \(≅\) \(ΔPQR\) by \(AAA\) congruence criterion.

Reasoning:

In this question, it is given that the angle measure of all the angles of triangle \(ABC\), \(∠A = 30^\circ\), \( ∠B = 40^\circ\) and \(∠C = 110^\circ\) is equal to the measure of all the angles of another triangle \(ΔPQR\), \(∠P = 30^\circ\), \(∠Q = 40^\circ\) and \(∠R =110^\circ\) and you can justify the congruence of \(ΔABC\) and \(ΔPQR\) by \(AAA\) criterion or not. You can justify your answer by using the property based on \(AAA\) congruence of two triangles. We know that there is no such thing as \(AAA\) congruence of two triangles: Two triangles with equal corresponding angles need not to be congruent. In such a correspondence, one of them can be an enlarged copy of the other (They would be congruent only if they are exact copies of one another).

Steps:

No, \(AAA\) property cannot justify \(ΔABC\) \(≅\) \(ΔPQR\) because, this property represents that these two triangles have their respective angles of equal measures, but it gives no information about their sides. Two triangles with equal corresponding angles need not to be congruent. In such a correspondence, one of them can be an enlarged copy of the other. Therefore, \(AAA\) does not prove that the two triangles \(ABC\) and \(PQR\) are congruent.