EXPLAINE congruency of triangles

 The triangle that congruent or equal in shape and size they were known as congruent

 i means they were known as congruency of triangles

 the triangles which are in same size,angle,shape is called congruence of triangles

Congruent Triangles

A polygon made of three line segments forming three angles is known as Triangle.

Two triangles are said to be congruent if their sides have the same length and angles have same measure. Thus two triangles can be superimposed side to side and angle to angle.

In the above figure, Δ ABC and Δ PQR are congruent triangles. This means,

Vertices:  A and P, B and Q, and C and R are same.

Sides:  AB=PQ, QR= BC and AC=PR;

Angles:  ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.

Congruent triangles are triangles having corresponding sides and angles to be equal. Congruence is denoted by the symbol ≅.  They have the same area and the same perimeter.

CPCT Rules in Maths

The full form of CPCT is Corresponding parts of Congruent triangles. Congruency can be predicted without actually measuring the sides and angles of a triangle. Different rules of congruency are as follows.

• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• RHS (Right angle-Hypotenuse-Side)

Let us learn them all in detail.

SSS (Side-Side-Side)

If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.

In the above-given figure, AB= PQ, QR= BC and AC=PR, hence Δ ABC ≅ Δ PQR.

SAS (Side-Angle-Side)

If any two sides and angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.

In above given figure, sides AB= PQ, AC=PR and angle between AC and AB equal to angle between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.

ASA (Angle-Side- Angle)

If any two angles and side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.

In above given figure, ∠ B = ∠ Q, ∠ C = ∠ R and sides between ∠B and ∠C , ∠Q and ∠ R are equal to each other i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.

AAS (Angle-Angle-Side)

AAS stands for Angle-angle-side. When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.

Students sometime may get confused AAS with ASA congruency. But remember that AAS is for non-included side whereas ASA is for included sides of the triangles.

If there are two triangles say ABC and DEF, then as per AAS rule:

∠B = ∠E

∠C = ∠F

AB = DE

Hence,

Δ ABC ≅ Δ DEF

RHS (Right angle- Hypotenuse-Side)

If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.

In above figure, hypotenuse XZ = RT and side YZ=ST, hence triangle XYZ ≅ triangle RST.

Solved Example

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