Question

In the following figure, the sides AB and BC and the median AD of triangle ABC are respectively equal to the sides PQ and QR and median PS of the triangle PQR. Prove that $\Delta$ PQR are congruent.

Answer 1

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 $\Delta$ ABC & PN is the median of $\Delta$ PQR.

AB = PQ,

BC = QR &

AM = PN

To Show:
(i) $\Delta$ ABM ≅ $\Delta$ PQN
(ii) $\Delta$ABC ≅ $\Delta$ PQR


Proof:
Since AM & PN is the median of $\Delta$ ABC


(i) $\frac{1}{2}BC=BM$
$\frac{1}{2}QR=QN$
(AM and PN are median)
Now,
BC = QR. (given)

⇒ $\frac{1}{2}BC=\frac{1}{2}QR$
(Divide both sides by 2)
⇒ BM = QN

In $\Delta$ ABM and $\Delta$ PQN,
AM = PN (Given)
AB = PQ (Given)
BM = QN (Proved above)

Therefore,
$\Delta$ ABM ≅ $\Delta$ PQN
(by SSS congruence rule)
$\angle B=\angle Q$ (CPCT)


(ii) In $\Delta$ ABC & $\Delta$ PQR,

AB = PQ (Given)
$\angle B=\angle Q$ (proved above in part i)
BC = QR (Given)
Therefore,
$\Delta$ ABC ≅ $\Delta$ PQR
( by SAS congruence rule)