TRIANGLE CONGRUENCE POSTULATES AND THEOREMS

Two triangles are said to be congruent, if they have the same shape and same size.

Following are the triangle congruence postulates and theorems :

SSS Congruence Postulate

Side-Side-Side (SSS) Congruence Postulate

Explanation :

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

SAS Congruence Postulate

Side-Angle-Side (SAS) Congruence Postulate

Explanation :

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

ASA Congruence Postulate

Angle-Side-Angle (ASA) Congruence Postulate

Explanation :

If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

AAS Congruence Postulate

Angle-Angle-Side (AAS) Congruence Postulate

Explanation :

If two angles and non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

HL Theorem

Hypotenuse-Leg (HL) Theorem

Explanation :

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

This principle is known as Hypotenuse-Leg theorem. 

LA Angle Theorem

Leg-Acute (LA) Angle Theorem

Explanation :

If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the two right triangles are congruent.

This principle is known as Leg-Acute Angle theorem.

HA Angle Theorem

Hypotenuse-Acute (HA) Angle Theorem

Explanation :

If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

This principle is known as Hypotenuse-Acute Angle theorem.

LL Theorem

Leg-Leg (LL) Theorem

Explanation :

If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.

This principle is known as Leg-Leg theorem.

Solved Problems

Problem 1 :

In the diagram given below, prove that ΔPQW ≅ ΔTSW.

Solution :

Problem 2 :

In the diagram given below, prove that ΔABC ≅ ΔFGH.

Solution :

Because AB = 5 in triangle ABC and FG = 5 in triangle FGH, 

AB ≅ FG

Because AC = 3 in triangle ABC and FH = 3 in triangle FGH,

AC ≅ FH

Use the distance formula to find the lengths of BC and GH.

Length of BC :

BC = √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = B(-7, 0) and (x₂, y₂) = C(-4, 5).

BC = √[(-4 + 7)² + (5 - 0)²]

=  √[3² + 5²]

= √[9 + 25]

= √34

Length of GH :

GH = √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = B(1, 2) and (x₂, y₂) = C(6, 5).

GH = √[(6 - 1)² + (5 - 2)²]

= √[5² + 3²]

= √[25 + 9]

= √34

Conclusion :

Because BC = √34 and GH = √34,

BC ≅ GH

All the three pairs of corresponding sides are congruent. By SSS congruence postulate,

ΔABC ≅ ΔFGH

Problem 3 :

In the diagram given below, prove that ΔAEB ≅ ΔDEC. 

Solution :

Problem 4 :

In the diagram given below, prove that ΔABD ≅ ΔEBC. 

Problem 5 :

In the diagram given below, prove that ΔEFG ≅ ΔJHG.

Solution :

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