Congruent Triangles

The word ‘congruent’ is used to describe objects that have the same shape or dimension. Congruence is the term used to define an object and its mirror image. Two or more objects are said to be congruent if they superimpose on each other or in other words they are of same shape and size. This property of being congruent is called congruency.

What are Congruent Triangles

Congruent triangles are triangles having all three sides of exactly the same length and all three angles of exactly the same measure. Thus, congruent triangles are mirror image of each other. On the other hand, triangles that are not congruent are called non-congruent triangles. In triangles, the abbreviation CPCT – Corresponding Parts of Congruent Triangles is used to show congruency.

Shown below are two triangles, ΔABC and ΔEFG that are congruent, which is mathematically represented as ΔABC ≅ ΔEFG. Their corresponding parts are shown below.

Corresponding Vertices	A and E, B and F, C and G
Corresponding Sides	AB ≅  EF, BC ≅  FG, AC ≅  EG
Corresponding Angles	∠A ≅ ∠E, ∠B ≅ ∠F, ∠C ≅ ∠G

Thus, the corresponding parts of congruent triangles are congruent.

Properties

Triangles can become congruent through flip (reflection), turn (rotation), or side-wise movement (translation) as shown below. It means, one triangle can be congruent to the other although their equal sides and angles are not in the same position. 

1) Reflexive Property

It states that the mirror image of any triangle is always congruent to it.

2) Symmetric Property

It states that the congruence works in both frontward and backward directions such that a triangle formed by rotation is always congruent to the other.

3) Transitive Property

It states that if two triangles are congruent to a third triangle, they are also congruent to each other.

Proving Triangles Congruency: Rules & Theorems

Triangle congruence is a set of rules or measures used to prove if two or more triangles are congruent. The five ways of identifying congruent triangles are shown below.

1) Side–Side-Side (SSS) Theorem

It states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

2) Side–Angle–Side (SAS) Theorem

It states that if two sides and the included angle of one triangle are congruent to two sides and included angle of another triangle, then the two triangles are congruent.

3) Angle–Side–Angle (ASA) Theorem

It states that if two angles and the included side of one triangle are congruent to two angles and included side of another triangle, then the two triangles are congruent.

4) Angle–Angle–Side (AAS) Theorem

It states that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

5) Hypotenuse-Leg (HL) Rule of a Right Angle Theorem

It states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

Theorems Not Applicable to Prove Triangles Congruency

AAA Rule: If all the corresponding angles of a triangle are congruent with the corresponding angles of another triangle, the triangles will be of the same shape but not necessarily of the same size. Thus, they will not be congruent.

SSA Rule: If two corresponding sides and an angle are congruent, the said triangles may not be congruent as there are two different shapes possible for the triangle. 

Congruent vs Similar Triangles

As we know, congruent triangles are triangles having the same shape and the same size. Mathematically, triangles with same side length and angle measure are called congruent triangles. In contrast, triangles having same shape but of different size are called similar triangles.

Thus, all congruent triangles are similar but all similar triangles are not congruent.

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  • HL Triangle