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 :

1. Side - Side - Side (SSS) Congruence Postulate

2. Side - Angle - Side (SAS) Congruence Postulate

3. Angle - Side - Angle (ASA) Congruence Postulate

4. Angle - Angle - Side (AAS) Congruence Postulate

5. Hypotenuse - Leg (HL) Theorem

6. Leg - Acute (LA) Angle Theorem

7. Hypotenuse - Acute (HA) Angle Theorem

8. Leg - Leg (LL) Theorem

Caution :

SSA and AAA can not be used to test congruent triangles.

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 :

Statements

PQ ≅ ST

PW ≅ TW

QW ≅ SW

ΔPQW ≅ ΔTSW

Reasons

Given

Given

Given

SSS Congruence Postulate

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 (x1y1) = B(-7, 0) and (x2y2) = 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 (x1y1) = B(1, 2) and (x2y2) = 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 :

Statements

AE ≅ DE, BE ≅ CE

∠1 ≅ ∠2

ΔAEB ≅ ΔDEC

Reasons

Given

Vertical Angles Theorem

SAS Congruence Postulate

Problem 4 :

In the diagram given below, prove that ΔABD ≅ ΔEBC

Statements

BD ≅ BC

AD || EC

∠D ≅ ∠C

∠ABD ≅ ∠EBC

ΔABD ≅ ΔEBC

Reasons

Given

Given

Alternate Interior Angles Theorem

Vertical Angles Theorem

ASA Congruence Postulate

Problem 5 :

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

Solution :

Statements

FE ≅ JH

∠E ≅ ∠J

∠EGF ≅ ∠JGH

ΔEFG ≅ ΔJHG

Reasons

Given

Given

Vertical Angles Theorem

AAS Congruence Postulate

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