Theorem 7.1 Class 9 Maths Explanation with Proof | Angle side angle (ASA)congruence rules

Theorem 7.1 Class 9 Maths Explanation with Proof

In this post, we will discuss theorem 7.1 class 9 Maths which is related to Chapter 7 Triangles Class 9 Mathematics. After understanding theorem 7.1, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

Theorem 7.1 Class 9 Proof

Here you can understand the theorem 7.1 class 9 with proof along with the video. 

Theorem 7.1 (ASA- Angle side angle) congruent rules Class 9 Maths

Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle.

Given 

Here we have been given two triangles ABC and PQR 

In ∠B=∠C, BC=QR, and ∠C=∠R

To prove: ∆ABC=∆PQR

Proof

There are three possibilities

Case 1 -When AB= PQ

Then in

 ∆ABC=∆PQR

AB=PQ

∠B=∠Q              (given)

BC=QR                (given)

So , ∆ABC=∆PQR (By SAS congruence rule)

Case 2  - When AB>PQ

Then take a point S on AB, such that SB=PQ

Now, in ∆SBCand ∆PQR

SB=PQ

∠B=∠Q                 (given)

And BC= QR         (given)

So ∆SBC=∆PQR(by SAS congruence rule)

∴∠SCB=∠PQR        (by CPCT)

But ∠ACB=∠PQR     (given)

So, ∠SCB=∠ACB

It is possible only if S coincides with A

Then SB=AB=PQ

So ∆ABC=∆PQR

Case 3: When  AB<PQ

Then take a point M on PQ, such that

MQ=AB

Now proceed same  as in case  2 to get

∠B=∠Q                      (given)

BC=QR                       (given)

∆ABC=∆MQR 

∠ACB=∠MRQ          (CPCT)

But ∠ACB=∠PRQ     (given)

So, ∠MRQ=∠PRQ

It is possible only if M coincides with P

∆ABC=∆PQR 

MQ=AB=PQ

∆ABC=∆PQR 

Hence in all three cases, we conclude that 

∆ABC=∆PQR 

Hope this Theorem 7.1 (ASA- Angle side angle) congruent rules Class 9 Maths will be useful for you. 

Related Topics

1. Exercise 7.1