Congruent Triangles Rules

This section covers congruent triangles rules and basics. But before that, let us understand some basic concepts of congruence.

Meaning of Congruence

We can use the concept of congruence when things, shapes or objects are identical.
In geometry, when objects and figures have same shape and size and they are mirror image of each other, then they are congruent.

Congruent Triangles

We know that a triangle has three sides and three angles.
Two triangles are congruent if their sides are of same length and they have equal angles. Also, if we superimpose one triangle on another it will completely cover the other triangle.
The congruency between two triangles is represented by ≅ .

 Understanding the concept of correspondence

Congruent triangles exampleLet us take two triangles ΔABC and ΔPQR.
Here as we can see, Sides
AB = PQ,
BC = QR
and AC = PR
As their sides are equal, we can say that they are congruent. 
i.e. ΔABC ≅ ΔPQR
We have to superimpose ΔPQR on ΔABC in such a way that the equal sides falls upon each other and ΔPQR completely hides ΔABC.
This is possible when PQ covers AB, QR covers BC and PR covers AC. Also ∠P covers ∠A, ∠Q covers ∠B and ∠R covers ∠C.

Clearly, this means we have to carefully understand which vertex of one triangle corresponds to which vertex of another triangle. In this case, vertex P corresponds to A, Q corresponds to B and R corresponds to C.
Thus, while denoting the congruency between these two triangles, we should keep in mind that the vertex corresponds to each other.
Hence, ΔABC ≅ ΔPQR.

Corresponding parts of congruent triangles

 When two triangles are congruent their corresponding parts are equal. We write this as CPCT.

Congruent Triangles Rules

Following are the congruent triangles rules that we are going to study.

  1. SSS (Side – Side – Side) Congruency
  2. SAS (Side – Angle – Side) Congruency
  3. ASA (Angle – Side – Angle) Congruency
  4. AAS (Angle – Angle – Side) Congruency
  5. RHS (Right angle – Hypotenuse – Side) Congruency
    congruent triangles

1. SSS (Side – Side – Side) Congruency

sss congruency

When the three sides of a triangle are equal to the other three sides of another triangle, then the triangles are said to be congruent by SSS congruency.
Here,
AB = PQ,
BC = QR , 
& AC = PR
Hence, ΔABC ≅ ΔPQR, by SSS congruency.

 

2. SAS (Side – Angle – Side) Congruency

sas congruency

When two sides and the included angle on one triangle is equal to the two sides and included angle of another triangle, then the triangles are congruent by SAS congruency.
Here,
AB = PQ,
∠B = ∠Q { Included angle between the equal sides}
& BC = QR
Hence, ΔABC ≅ ΔPQR, by SAS congruency.

3. ASA (Angle – Side – Angle) Congruency

ASA congruency

When two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are said to be congruent by ASA congruency.
Here, 
∠A = ∠P,
AB = PQ, { Included side between equal angles}
& ∠B = ∠Q
Hence, ΔABC ≅ ΔPQR, by ASA congruency.

4. AAS (Angle – Side – Angle) Congruency

AAS congruency

When two angles and the non-included side of one triangle are equal to two angles and the non- included side of another triangle, then the triangles are said to be congruent by AAS congruency.
Here,
∠A = ∠P,
∠C = ∠R,
& AB = PQ 
Hence, ΔABC ≅ ΔPQR, by AAS congruency.

5. RHS (Right Angle – Hypotenuse – Side) Congruency

RHS congruency

This congruency is applicable on Right angled Triangles.
When hypotenuse and one side of a right triangle is equal to the hypotenuse and other side of another right triangle, then the triangles are said to be congruent by RHS congruency.
Here, 
AC = PR { Hypotenuse)
∠B = ∠Q = 90º
BC = QR,
Hence,ΔABC ≅ ΔPQR, by RHS congruency.

You may also want to learn the following – 

Triangles basic properties
Types of angles