SAS Congruency Criteria

To prove two triangles congruent,

We use SAS Criteria – Side Angle Side

In SAS Congruency Criteria,

• 2 sides of both the triangles are equal
• 1 angle between these side of both the triangles are equal.

For example

Here,

  2 sides and the angle between  them are equal.

Let’s do some examples

Are these triangles congruent?

In ∆ABC and ∆FED,

BC = ED      (Both are 10 cm )

∠C = ∠D      (Both are 45 °)

AC = FD      ( Both are 8 cm )

∴ ∆ABC ≅ ∆FED     (SAS Congruence Rule)

Here,

C ⟷ D

A ⟷ F

B ⟷ E

Are these triangles congruent?

In ∆PQR and ∆UTS,

PR = US     (Both are 20 cm )

∠R = ∠S     (Both are 15 °)

QR = TS     (Both are 10 cm )

∴ ∆PQR ≅ ∆UTS  (SAS Congruence Rule)

Here,

R ⟷ S

P ⟷ U

Q ⟷ T

Are these triangles congruent?

In ∆ABC and ∆RQP,

AC = RP  ( Both are 2.5 cm )

∠C = ∠P  ( Both are 35 °)

BC = QP  ( Both are 3 cm )

∴ ∆ABC ≅ ∆RQP ( SAS Congruence Rule )

Here,

C ⟷ P

A ⟷ R

B ⟷ Q

Are these triangles congruent?

In ∆DEF and ∆RPQ,

EF = PQ  (Both are 3 cm )

∠F = ∠Q  (Both are 40 °)

DF = RQ  (Both are 3.5 cm )

∴ ∆DEF ≅ ∆RPQ  (SAS Congruence Rule )

Here,

F ⟷ Q

D ⟷ R

E ⟷ P

Are these triangles congruent?

In ∆PRS and ∆RPQ,

RS = PQ     ( Both are 3.5 cm )

∠R = ∠P     (Both are 30 °)

PR = RP    (Common )

∴ ∆PRS ≅ ∆RPQ  ( SAS Congruence Rule)

Here,

P ⟷ R

S ⟷ Q

R ⟷ P

In the following diagram:-

In Fig, AB and CD bisect each other at O.

(i) State the three pairs of equal parts in two triangles AOC and BOD.

(ii) Which of the following statements are true?

(a) ∆AOC ≅ ∆DOB

(b) ∆AOC ≅ ∆BOD

In ∆AOC and ∆BOD,

OA = OB                      (Given )

 ∠AOC = ∠BOD          (Vertically opposite angles)  

OC = OD                    ( Given)

  ∴ ∆AOC ≅ ∆BOD      ( SAS Congruence Rule )

Here,

A ⟷ B

O ⟷ O

C ⟷ D