Question

Que: Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig. 7.40). Show that:

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR​

Answer 1

Solution:

Given;

AB = PQ,

BC = QR and

AM = PN

(i) 1/2 BC = BM and 1/2QR = QN (Since AM and PN are medians)

Also, BC = QR

So, 1/2 BC = 1/2QR

⇒ BM = QN

In ΔABM and ΔPQN,

AM = PN and AB = PQ (Given)

BM = QN (Already proved)

∴ ΔABM ≅ ΔPQN by SSS congruency.

(ii) In ΔABC and ΔPQR,

AB = PQ and BC = QR (Given)

∠ABC = ∠PQR (by CPCT)

So, ΔABC ≅ ΔPQR by SAS congruency.

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Hope it will be helpful :)

Answer 2

Solution::

Congruence of triangles:

Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:

There are 4 criteria for congruence of triangles.

SAS( side angle side):

Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.

SSS(side side side):

Three sides of One triangle are equal to the three sides of another triangle then the two Triangles are congruent.

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We know that median bisects opposite side. Use this property and then show that given parts by using SSS and SAS congruence rule.

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Solution:

Given:

AM is the median of ∆ABC & PN is the median of ∆PQR.

AB = PQ, BC = QR & AM = PN

To Show:

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR

Proof:

Since AM & PN is the median of ∆ABC

(i) 1/2 BC = BM &

1/2QR = QN

(AM and PN are median)

Now,

BC = QR. (given)

⇒ 1/2 BC = 1/2QR

(Divide both sides by 2)

⇒ BM = QN

In ΔABM and ΔPQN,

AM = PN (Given)

AB = PQ (Given)

BM = QN (Proved above)

Therefore,

ΔABM ≅ ΔPQN

(by SSS congruence rule)

∠B = ∠Q (CPCT)

(ii) In ΔABC & ΔPQR,

AB = PQ (Given)

∠B = ∠Q(proved above in part i)

BC = QR (Given)

Therefore,

ΔABC ≅ ΔPQR

( by SAS congruence rule)

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Hope this will help you.....