Congruence of Triangles: Definition, Properties, Rules for Congruence

Congruence of Triangles: The congruence of a triangle depends upon the measurements of sides and angles of the two triangles. There are a few criteria, based on which it can be it can be decided whether two given triangles are congruent or not.

The congruence of triangles is used to define the given triangle and its mirror image. Two objects are said to be congruent if the given object and its mirror image superimpose each other. In this article, we will understand the congruence of triangles, properties, and types of congruences.

What is Congruence of Triangles?

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

Two triangles are congruent if all the three corresponding sides of two triangles are the same. There are a few more criteria also. Thus, congruent triangles are mirror images to each other. The mathematical symbol represents congruence is \( \cong .\)

In the triangles \(\Delta ABC,\Delta EFG\) as shown below, we can identify corresponding angles and corresponding sides are equal.

Criteria for Congruence of Triangles

Two triangles of the same size and shape are called congruent triangles. When we rotate, reflect, or translate a triangle, its position or appearance seems identical to the other, also called congruent.

Based on the experiments, there are mainly \(5\) conditions or rules to compare the two triangles to be congruent. They are \(SSS,SAS,ASA,AAS,\) and \(RHS\) congruence properties.

SSS (Side – Side – Side) Congruence

If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

SSS Criterion stands for Side-Side-Side Criterion.

In the above-given triangles \(\Delta ABC,\Delta EFG,\)
1.\(AC = EG\) (Corresponding equal sides shown by a single line).
2. \(BC = FG\) (Corresponding equal sides shown by two lines).
3. \(AB = EF\) (Corresponding equal angles)

Here, sides \(\left({AB,BC,AC} \right)\) of triangle \(ABC\) are equals to the corresponding sides \(\left({EF,FG, GE} \right)\) of the triangle \(EFG.\)

Hence, the given triangles are congruent to each other.
\(\Delta ABC \cong \Delta EFG\)

SAS (Side – Angle – Side) Congruence

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.

SAS Criterion stands for Side-Angle-Side Criterion.

In the above-given triangles \(\Delta ABC,\,\Delta EFG,\)
1. \(AC = EG\) (Corresponding equal sides shown by a single line).
2. \(AB = EF\) (Corresponding equal sides shown by two lines).
3. \(\angle BAC = \angle FEG\) (Corresponding equal angles)

Here, sides \(\left({AB,AC} \right)\) and angle \(\angle BAC\) of triangle \(ABC\) are equals to the corresponding sides \(\left({EF,GE}\right)\) and corresponding angle \(\angle FEG\) of the triangle \(EFG.\)

Hence, the given triangles are congruent to each other.
\(\Delta ABC \cong \Delta EFG\)

ASA (Angle – Side – Angle) Congruence

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.

ASA Criterion stands for Angle – Side – Angle Criterion.

In the above-given triangles \(\Delta ABC,\Delta EFG,\)
1. \(BC = FG\) (Corresponding equal sides shown by a single line).
2. \(\angle ACB = \angle EGF\) (Corresponding equal angles shown by a single arc)
3. \(\angle ABC = \angle EFG\) (Corresponding equal angles shown by two arcs)

Here, the side \(\left({BC}\right)\) and angles\(\left({\angle ACB,\angle ABC} \right)\) of triangle \(ABC\) are equals to the corresponding side \(\left({FG} \right)\) and corresponding angles \(\left({\angle EGF,\angle EFG} \right)\) of the triangle \(EFG.\)

Hence, the given triangles are congruent to each other.
\(\Delta ABC \cong \Delta EFG\)

AAS (Angle – Angle – Side) Congruence

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.

AAS Criterion stands for Angle – Angle – Side Criterion.

In the above-given triangles \(\Delta ABC,\Delta EFG,\)
1. \(AB = EF\)(Corresponding equal sides shown by a single line).
2. \(\angle CAB = \angle GEF\) angles (corresponding equal angles shown by a single arc)
3. \(\angle ACB = \angle EGF\) angles (corresponding equal angles shown by two arc

Here, the side \(\left({AB} \right)\) and angles \(\left({\angle ACB,\angle CAB} \right)\) of triangle \(ABC\) are equals to the corresponding side \(\left({FG} \right)\) and corresponding angles \(\left({\angle EGF,\angle GEF} \right)\) of the triangle \(EFG.\)

Hence, the given triangles are congruent to each other.
\(\Delta ABC \cong \Delta EFG\)

RHS (Right Angle – Hypotenuse – Side) Congruence

It states that If the hypotenuse and a side of a right-angled triangle are equivalent to the hypotenuse and a side of the second right-angled triangle, then the two right triangles are congruent.

RHS Criterion stands for Right Angle – Hypotenuse – Side Criterion.

In the above-given triangles \(\Delta ABC,\Delta EFG,\)
1. \(AC = EF\) (Corresponding equal sides shown by a single line).
2. \(BC = FG\) (Corresponding equal sides shown by two lines).
3. \(\angle ABC = \angle EGF\)(Right angles)

Here, sides \(\left({BC,AC} \right)\) and right angle \(\angle ABC\) of triangle \(ABC\) are equals to the corresponding sides \(\left({EF,GF} \right)\) and corresponding right angle \(\angle FGE\) of the triangle \(EFG.\)

Hence, the given triangles are congruent to each other.
\(\Delta ABC \cong \Delta EGF\)

Properties of Congruence of Triangles

Some of the properties of the congruence of triangles are:

Reflexive Property

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

Symmetric Property

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

Transitive Property

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

Congruence and Similarity of Triangles

As in both similarity and congruence, the corresponding angles of both the triangles are equal.

Solved Examples – Congruence of Triangles

Q.1. Check if the given triangles are congruent or not, as mentioned below.

Ans: In the given triangles \(\Delta ABC,\Delta PQR\)
\(\angle ABC = \angle PQR = {90^ \circ }\)
\(AC = PR = 5 \ {\text{cm}},\) (Hypotenuse of the given triangles)
\(QP = BC = 4 \ {\text{cm}},\) (Side of the given triangles)
Hence, by the \(R.H.S\) rule of congruence, given triangles are congruent.
\(\Delta ABC \cong \Delta RQP\)

Q.2. In the below figure, \(OA = OB\) and \(OC = OD,\) then show that \(\Delta AOD \cong \Delta BOC.\)

Ans: Observe the triangles \(\Delta AOD\) and \(\Delta BOC,\)
\(OA = OB\) (Given)
\(OC = OD\)(Given)
Also, since \(\angle AOD\) and \(\angle BOC\) form a pair of vertically opposite angles, we have
\(\angle AOD = \angle BOC.\)
Here, sides and included angles of ∆AOD equal to the corresponding sides and angle of \(\Delta BOC.\) By \(S.A.S\) congruence rule, \(\Delta AOD \cong \Delta BOC.\)

Q.3. \(AB\) is a line segment, and line \(l\) is its perpendicular bisector. If a point P lies on \(l\), show that \(P\) is equidistant from \(A\) and \(B.\)

Ans: Line \( \bot AB\) and passes through \(C,\) which is the mid-point of \(AB.\)
We have to show that \(PA = PB.\)
Consider \(\Delta PCA\) and \(\Delta PCB.\) We have
\(AC = BC\) (\(C\) is the mid-point of \(AB\)) \(\angle PCA = \angle PCB = {90^ \circ }\) (Given)
\(PC = PC\) (Common)
By using the \(S.A.S\) rule, \(\Delta PCA \cong \Delta PCB\)
We know that the corresponding sides of the congruent triangles are equal.
\(PA = PB\)

Q.4. Line \(l\) is the bisector of an angle \(\angle A\) and \(\angle B\) is any point on \(l.BP\) and \(BQ\) are perpendiculars from \(B\) to the arms of \(\angle A.\) Show that \(\Delta APB \cong AQB.\)

Ans: Given \(l\) is the bisector of \(\angle A.\)
In \(\Delta APB\) and \(\Delta AQB,\) \(\angle QAB = \angle PAB\) (Given)
\(\angle AQB = \angle APB = {90^ \circ }\) (Given)
\(AB = AB\) (Common)
By using \(A.A.S\) congruence rule,
\(\Delta APB \cong \Delta AQB\)

Q.5. \(P\) is a point equidistant from two lines \(l\) and \(m\) that intersect at point \(A.\) Show that the line \(AP\) bisects the angle between them.

Ans: Let PB \(PB \bot l,\,PC \bot m.\)
It is given that\(PB = PC.\)
We need to show that \(\angle PAB = PAC.\)
In \(\Delta PAB\) and \(\Delta PAC,\)
\(PB = PA\) (Given)
\(\angle PBA = \angle PCA = {90^ \circ }\) (Given)
\(PA = PA\) (Common)
So, \(\Delta PAB \cong \Delta PAC\) (\(RHS\) rule)
\(\angle PAB = \angle PAC\left({CPCT} \right)\)

Summary

In this article, we have studied the congruence of triangles and the rules of triangles such as side-side-side rule \(\left({SSS} \right),\) side-angle-side rule\(\left({SAS} \right),\) angle-side-angle rule \(\left({ASA} \right),\) angle-angle-side rule \(\left({AAS} \right),\) right angle-hypotenuse-side rule \(\left({RHS} \right).\)

Two triangles are congruent if one or more of the above criteria is/are fulfilled.

Frequently Asked Questions(FAQ) – Congruence of Triangles

Q.1. How to prove the congruence of triangles?
Ans: The congruence of triangles is proved by using the congruent rules as given below:
1. \(S.A.S\left({Side – Angle – Side} \right)\)
2. \(S.S.S\left({Side – Side – Side} \right)\)
3. \(A.S.A\left({Angle – Side – Angle} \right)\)
4. \(A.A.S\left({Angle – Angle – Side} \right)\)
5. \(RHS\left({Right\,Angle – Hypotenuse – side} \right)\)

Q.2. What is the congruence of triangles?
Ans: The different criteria for congruence of two triangles are the side-side-side rule \(\left({SSS} \right),\) side-angle-side rule \(\left({SAS} \right),\) angle-side-angle rule \(\left({ASA} \right),\) angle-angle-side rule \(\left({AAS} \right),\) right angle-hypotenuse-side rule \(\left({RHS} \right).\)

So, two triangles are congruent if one or more of the above criteria is/are fulfilled.

Q.3. How can you identify the congruence in triangles?
Ans: If we identify the parts of one triangle equal to the corresponding parts of the other triangle, we can identify the congruence in triangles.

Q.4. What are congruent triangles?
Ans: Two triangles are congruent if they are exact copies of each other, and when superimposed, they cover each other exactly. Two congruent triangles have the same perimeter and area. Q.3. Is AAA a congruence rule?
Ans: By this rule, if all the corresponding angles of a triangle measure equal, the triangles will have the same shape but not necessarily the same size. So, it will be a case of two triangles of the same shape, but one is bigger than the other. Hence, \(AAA\) is not a congruence rule.