Congruence | Congruence of two Triangles

In geometry, two figures are congruent if they have the same shape and size. Symbol of congruence is

Congruence of two line Segment:

Two line segment are congruence if and only if their length are equal.

Congruence Line | Congruence of Triangle

In above figure,
PQ = RS

Congruence of two angle:

Two angle are congruence if and only if their measures are equal.

Congruence Angle | Congruence of Triangle

In above figure,
∠B = ∠E

Congruence of two Triangles:

Two triangles are congruence if and only if there exists a corresponding between their vertices such that the corresponding angles of two triangles are equal.

Congruence of Triangle

Let’s take two triangles If Δ ABC and Δ DEF.
In above figure,
AB = DE & BC = EF & AC = DF
∠A = ∠D & ∠B = ∠E & ∠C = ∠F
so Δ ABC and Δ DEF is said to be congruent to each other and are written as Δ ABC ≅ Δ DEF.

Rules of Congruence of two Triangles:

Different rules of congruency are as follows:

SSS (Side-Side-Side):

If the three sides of one triangle are equal to the corresponding three sides of the second triangle, then the two triangles are said to be SSS Congruence of two Triangles.

SSS Congruence of Triangle

In above figure,
AB = DE & BC = EF & AC = DF
∠A = ∠D & ∠B = ∠E & ∠C = ∠F

SAS (Side-Angle-Side):

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the second triangle, then the two triangles are said to be SAS Congruence of two Triangles.

SAS-Congruence of twoTriangle

In above figure,
AB = DE & BC = EF
∠B = ∠E

ASA (Angle-Side-Angle):

If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the second triangle, then the two triangles are said to be ASA Congruence of two Triangles.

ASA-Congruence of two Triangle

In above figure,
AC = DF
∠A = ∠D & ∠C = ∠F

AAS (or SAA) (Angle-Angle-Side or Side-Angle-Angle):

If two angles and the non-included side of one triangle are equal to the corresponding two angles and the non-included side of the second triangle, then the two triangles are said to be ASA Congruence of two Triangles.

AAS - Congruence of two Triangles

RHS (Right angle- Hypotenuse-Side):

If the hypotenuse and side of one right triangle are equal to the corresponding hypotenuse and side of the second triangle, then the two triangles are said to be RHS Congruence of two Triangles.

RHS-Congruence of two Triangles

CPCT

This CPCT stands for Corresponding Parts of Congruent Triangles are Congruent an abbreviated version of the definition of congruent triangles.

If Δ ABC and Δ DEF are congruent, that is,
Δ ABC ≅ Δ DEF

with corresponding pairs of angles at vertices A and D; B and E; and C and F, and with corresponding pairs of sides AB and DE; BC and EF; and CA and FD, then the following statements are true:

AB ≅ DE
BC ≅ EF
AC ≅ DF
∠BAC ≅ ∠EDF
∠ABC ≅ ∠DEF
∠BCA ≅ ∠EFD.

Point to Remember:

Congruent triangles coincides by superposition.
Congruent triangles are similar but the converse is not always true.

Related Topic:

Triangle (An Introduction), Types & Area of triangle

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