Criteria For Congruence Of Triangles

1. SSS Criterion (Side-Side-Side)

This criterion says that the two triangles will be congruent if their corresponding sides are equal.

If Side AB = DE

Side BC = EF

Side AC = DF

Then, ∆ABC ≅ ∆DEF

Example

In the two given triangles, ∆ABC and ∆DEF AB = 7 cm, BC = 5 cm, AC = 9 cm, DE = 7 cm, DF = 9 cm and EF = 5 cm. Check whether the two triangles are congruent or not.

Solution

In ∆ABC and ∆DEF,

 AB = DE = 7 cm,

BC = EF = 5 cm,

AC = DF = 9 cm

This show that all the three sides of ∆ABC are equal to the sides of ∆DEF.

Hence with the SSS criterion of congruence, the two triangles are congruent.

∆ABC ≅ ∆DEF

2. SAS Criterion (Side-Angle-Side)

This criterion says that the two triangles will be congruent if their corresponding two sides and one included angle are equal.

If Side AB = DE

Angle ∠B = ∠E

Side BC = EF

Then, ∆ABC ≅ ∆DEF

Example

In ∆JKN, JK = KN and AK is the bisector of ∠JKN, then

1. Find the three pairs of equal parts in triangles JKA and AKN.

2. Is ΔJKA ≅ ΔNKA? Give reasons.

Is ∠J = ∠N? Give reasons.

Solution

1. The three pairs of equal parts are:

JK = KN (Given)

∠JKA = ∠NKA (KA bisects ∠JKN)

AK = AK (common)

2. Yes, ΔJKA ≅ ΔNKA (By SAS congruence rule)

3. ∠J = ∠N(Corresponding parts of congruent triangles)

3. ASA criterion (Angle-Side-Angle)

This criterion says that the two triangles are congruent if the two adjacent angles and one included side of one triangle are equal to the corresponding angles and one included side of another triangle.

If Angle ∠B = ∠B’

Side BC = EF

Angle ∠C = ∠C’

Then, ∆ABC ≅ ∆A’B’C’

Example

In ∆LMN and ∆OPN, if  LMN = ∆NPO = 60°, LNM = 35° and LM = PO = 4 cm. Then check whether the triangle LMN is congruent to triangle PON or not.

Solution

In the two triangles ∆LMN and ∆OPN,

Given,

LMN = NPO = 60°

LNM = ∠PNO = 35° (vertically opposite angles)

So, ∠L of ΔLMN = 180° – (60° + 35°)   = 85° (by angle sum property of a triangle) similarly,

∠O of ΔOPN =180° – (60° + 35°) = 85°

Thus, we have ∠L = ∠O, LM = PO and ∠M = ∠P

 Now, side LM is between ∠L and ∠M and side PO is between ∠P and ∠O.

Hence, by ASA congruence rule,

∆LMN ≅ ∆OPN.

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