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Congruence of Triangles By SAS Criteria

Triangles I - Concepts

Calculating Perimeter of Triangle

Congruence of Triangles

Congruence of Triangles By SAS Criteria

Congruence of Triangles By ASA Criteria

Congruence of Triangles By AAS Criteria

Class - 9th IMO Subjects

Concept Explanation

Congruence of Triangles By SAS Criteria

Theorem: Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle.

Given: Two triangles ABC and DEF such that $large AB=DE, AC=DF;and;angle A=angle D$

To Prove: $large Delta ABCcong Delta DEF$

Proof: Place $large Delta ABC;over; Delta DEF$ such that vertex A falls on vertex D and the side AB falls on side DE,

As AB = DE [Given]

So vertex B falls on vertex E.

Since $angle A=angle D$ . Therefore, AC will fall on DF.

But AC = DF [ Given]

Therefore, C will fall on F.

Thus, AC coincides with DF.

Now, B falls on E and C falls on F. Therefore, BC coincides with EF.

Thus, $large Delta ABC$ will coincide with $large Delta DEF$ .

Hence, by definition of congruence, $large Delta ABCcong Delta DEF$

Illustration: In $large Delta ABC$ and $large Delta PQR$ , AB = PQ, BC = QR and CB and RQ are extended to X and Y respectively and $large angle ABX=angle PQY.$ Prove that $large Delta ABCcong Delta PQR.$

Solution: We have, $large angle ABX=angle PQY$

$large Rightarrow ;;;180^{circ}-angle ABC=180^{circ}-angle PQR$

[ $large because angle ABX+angle ABC=180$ $large therefore angle ABX=180^{circ}-angle ABC$ Similarly, $large angle POY=180^{circ}-angle PQR$ ]

$large Rightarrow ;;;-angle ABC=-angle PQR$

$large Rightarrow ;;;angle ABC=angle PQR$ .....(i)

In $large Delta ABC$ and $large Delta PQR,$ we have

AB = PQ [Given]

$large angle ABC=angle PQR$ [From (i)]

and, BC = QR [Given]

So, by SAS criterion of congruence, we have

$large Delta ABCcong Delta PQR$

Illustration: In the given figure, AC = AE, AB = AD and $angle BAD=angle EAC$ .Show that BC = DE

Solution: We have,

$angle BAD=angle EAC$ [Given]

$Rightarrow$ $angle BAD+angle DAC=angle EAC + angle DAC$ [Adding $angle DAC$ to both sides]

$Rightarrow$ $angle BAC=angle DAE$

Now, in triangles ABC and ADE,

$AB=AD$ [Given]

$angle BAC=angle DAE$ [From (i)]

and, $AC=AE$ [Given]

So, by SAS congruence criterion, we have

$Delta ABCcong Delta ADE$

$Rightarrow$ $BC=DE$ [CPCT]

Sample Questions

(More Questions for each concept available in Login)

Question : 1

In the figure shown above, if the two triangles are congruent by SAS criteria then which of the following conditions are not true?
A. AB = DE	B. AC = DF	C. BC = EF	D. $angle BAC=angle EDF$
Right Option : C
View Explanation

Question : 2

In the given figure, the congruency rule used in proving $Delta ACD cong Delta ADB$ is _________________
A. ASA	B. RHS	C. SSS	D. SAS
Right Option : D
View Explanation

Question : 3

In the figure given above, AC = CE and BC = CD and $angle ACB=angle DCE$ then $bigtriangleup ACBcong bigtriangleup DCE$ by which criterion?
A. SAS	B. AAA	C. ASA	D. AAS
Right Option : A
View Explanation