 # Congruence of Triangles

Congruence of Triangles: Congruence is the term used to define an object and its mirror image. Two objects or shapes are said to be congruent if they superimpose on each other. Their shape and dimensions are the same. In the case of geometric figures, line segments with the same length are congruent and angle with same measure are congruent. CPCT is the term, we come across when we learn about the congruent triangle. Let’s see the condition for triangles to be congruent with proof.

## 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, D ABC and D 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:  Ð A = Ð P, ÐB = ÐQ and Ð C= Ð 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 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)

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 D ABC ≅ D 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. Ð A = Ð P, hence D ABC ≅ D 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, Ð B = Ð Q, Ð C = Ð R and sides between Ð B and ÐC , Ð Q and Ð R are equal to each other i.e. BC= QR, hence D ABC ≅ D PQR.

### 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, the 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.

### Example

Let’s Work Out:

#### Example: In the following figure, AB = BC and AD = CD. Show that BD bisects AC at right angles. Solution: We are required to prove ∠BEA = ∠BEC = 90° and AE = EC.

Consider ∆ABD and ∆CBD,

AB = BC                                                (Given)

BD = BD                                                (Common)

Therefore, ∆ABD ≅ ∆CBD                      (By SSS congruency)

∠ABD = ∠CBD                                      (CPCTC)

Now, consider ∆ABE and ∆CBE,

AB = BC                                                (Given)

∠ABD = ∠CBD                                      (Proved above)

BE = BE                                                (Common)

Therefore, ∆ABE≅ ∆CBE                       (By SAS congruency)

∠BEA = ∠BEC                                      (CPCTC)

And ∠BEA +∠BEC = 180°                      (Linear pair)

2∠BEA = 180°                                       (∠BEA = ∠BEC)

∠BEA = 180°/2 = 90° = ∠BEC

AE = EC                                                (CPCTC)

Hence, BD is perpendicular bisector of AC.

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### What are Congruent Triangles?

Two triangles are said to be congruent if the three sides and the three angles of both the angles are equal in any orientation.

### What is the Full Form of CPCT?

CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.

### What are the Rules of Congruency?

There are 4 main rules of congruency for triangles:

• SSS Criterion: Side-Side-Side
• SAS Criterion: Side-Angle-Side
• ASA Criterion: Angle-Side- Angle
• RHS Criterion: Right angle- Hypotenuse-Side