Congruency of Objects: Definition, Congruent vs Similar, Examples

Congruency of objects: The word congruent refers to similar or twin. An object is congruent to another object when they are exactly the same in shape and size. Two objects are congruent when they are the mirror image of each other. Many theorems are connected with equality and congruency. We say two lines are congruent only if they have equal length, two angles are congruent if they have an equal measure, and two triangles are congruent if they have corresponding parts equal.

Congruent Definition in Geometry

Two objects or figures are said to be congruent if their corresponding parts are exactly the same and equal. And the two figures are said to have congruency.

Real-Life Examples: Some of the real-life examples are given below with figure:

1. Biscuits from the same packet

2. Pair of a die of the same board

3. \(1\) rupee coins

4. All the pages in a book

Congruent Objects

Congruent objects are identical in shape and size. Congruent figures have congruency. It is simple enough, but congruent objects need not face the same direction or be turned the same way to be congruent.

We observe that the two apple’s sizes, shape, colour, and details are exactly the same from the above figure. In geometry, certainly, we ignore the colour. We concentrate only on size and shape. Suppose we turn one apple on its side. They are still congruent, even though one is rotated. Rotation does not obstruct the congruency.

Drawing Congruent Figures

Take a sheet of paper and draw two exact same figures. Cut the two figures and now place them one over the other. We find that the figures are overlapping each other.

In geometry from the above figure, we can say that figure \(\left(F_{1}\right)\) is congruent to figure \(\left(F_{2}\right)\), which we write down as figure \({F}_{1} \cong\) figure \({F}_{2}\).

Properties of Congruence

The properties of congruence are discussed below:

1. Reflexive property: In congruency, an object or a figure is congruent to itself. For example, \(\angle P \cong \angle P\) for any angle \(P\).
2. Symmetric property: In congruency, the symmetric property means two figures are congruent to each other. For any two angles \(P\) and \(Q, \angle P \cong \angle Q\), then \(\angle Q \cong \angle P\).
3. Transitive property: Transitive property of congruence states that if two shapes are congruent to the third, then the first two shapes are congruent. If \(A \cong B\) and \(A \cong C\), then \(B \cong C\).

Transformations in Geometry:

In geometry transformations, shapes can be manipulated in four different ways:

1. Rotate (turn): If two figures are congruent, rotating one or both at different angles does not affect their congruency.
2. Translate (slide): Congruency of two figures is retained even when the position of the objects changes.
3. Reflect (flip): When one figure is the mirror image of another, the two figures are congruent. In other words, if you flip the image, the congruency remains.
4. Dilate (enlarge or shrink): Only this transformation will not let the congruency remain. When an object is shrunk or enlarged, the object is no more congruent.

For the first three ways, congruent figures remain congruent. When dilated, the congruency dissolves.

Condition for Congruency of Figures

Congruent Lines Segments: For two line segments to be congruent, they have equal or the same length. Below are given two line segments \(A B\) and \(C D\) with length \(8\) units each.

Take two line segments and place them on top of each other. So, two line segments are congruent if they are equal or the same.

Congruent Angles: For two angles to be congruent, their angle measurement should be equal. If angle \(A\) and angle \(B\) are \(45^{\circ}\) each, they are congruent.

Congruence of Triangles: For two triangles to be congruent, their sides should have equal lengths and angles, and they cover each other when kept one over the other.

In the above figure, \(\triangle A B C\) and \(\triangle P Q R\) are congruent triangles if the corresponding angles and corresponding sides in both the triangles are equal. Vertices: \(A\) and \(P, B\) and \(Q\), and \(C\) and \(R\) are the same.
Sides: \(A B=P Q, B C=Q R\) and \(A C=P R\)
Angles: \(\angle A=\angle P, \angle B=\angle Q\), and \(\angle C=\angle R\).
Thus, \(\triangle A B C \cong \triangle P Q R\)
Note: The following are the criteria for two triangles to be congruent.
1. SSS (side, side, side) 
2. SAS (side, angle, side) 
3. ASA (angle, side, angle)
4. AAS (angle, angle, side)
5. RHS (right angle-hypotenuse-side)

Congruent Shapes

Any two shapes can be congruent, whether it is square, star, polygon, etc. If the two figures are placed one over the other, they must coincide or completely cover each other. They must have an exact match even when they are flipped or rotated.

The following figure shows the congruent figures of different shapes:

Difference Between Congruent Objects and Similar Objects

Solved Examples – Congruency of Objects

Q.1. Prove that the given two figures are congruent.

Ans: In the given figure, the angles \(B\) and \(Y\) are \(40^{\circ}\) each, equal. We know that for two angles to be congruent, their angle measure must be equal. Thus, the given two angles or figures are congruent.

Q.2. Here are two triangles that you have to prove as congruent.

Ans: From the given figure,
In \(\Delta A B C\) and \(\Delta P Q R\),
\(\angle A=\angle P\) (given)
\(A B=P R\) (given)
\(A C=P Q\) (given)
By SAS criterion of congruency, \(\triangle A B C \cong \triangle P R Q\).

Q.3. Given below are the two congruent quadrilaterals.

Which angle in quadrilateral \(A B C D\) corresponds to \(\angle Q R S\) in quadrilateral \(P Q R S\) ?
Ans: We need to identify the corresponding parts in both quadrilaterals. We observe that \(\angle Q R S\) is marked with three arcs.
\(\angle C D A\) is also marked with three arcs.
This shows that \(\angle Q R S\) coincides with \(\angle C D A\). Thus, \(\angle C D A\) corresponds to \(\angle Q R S\).

Q.4. Malini is doing craftwork. She has four squares with given sides, square \(A=5 \mathrm{~cm}\), square \(B=7 \mathrm{~cm}\), square \(C=5 \mathrm{~cm}\), square \(D=8 \mathrm{~cm}\) She wants two squares that can coincide exactly when placed one over the other. Can you help her choose?
Ans: For two squares to coincide exactly, the condition is they must be congruent. Squares with the same sides will superimpose on each other. So, Malini should find two squares whose side lengths are exactly the same. In the given list, we can see that square \(A\) and Square \(C\) are having the same side length that is \(5 \mathrm{~cm}\). She can take square \(A\) and \(C\) as they can be placed exactly one over the other.

Q.5. If \(\angle R O T \cong \angle T O S \cong \angle S O P\), then find the angle which is congruent to \(\angle S O R\).

Ans: Given, \(\angle R O T \cong \angle T O S \cong \angle S O P\)
Let us consider, \(\angle R O T \cong \angle S O P\)
Adding \(\angle T O S\) on both sides, we get,
\(\Rightarrow \angle R O T+\angle T O S \cong \angle S O P+\angle T O S\)
\(\Rightarrow \angle S O R \cong \angle T O P\)
Thus, the angle congruent to \(\angle S O R\) is \(\angle T O P\).

Summary

In this article, we have discussed the congruency of objects. We learnt the definition of congruency of objects and examples with figures to visualise and understand. We learnt some of the properties that make two objects congruent, and we discussed transformation in geometry.

We also learnt how to draw congruent figures and conditions such as line segments, angles, triangles, and other figures. The last topic is the difference between congruent figures and similar figures. Some examples have been solved to understand the concept clearly.

Congruence and Congruence of Plane Figures

Frequently Asked Questions (FAQs)

Q.1. What is congruence in shapes?
Ans: Congruent shapes have the same size and the same shape. In other words, if you keep a figure in front of a mirror, the image that you see is congruent or ” equal ” to the object.

Q.2. Can circles be congruent?
Ans: Yes, circles can be congruent. By definition, all radii of a circle are of equal length as all the points on a circle are of the same distance from the centre. All circles have a diameter, too. Thus, the circle of the same radii or same diameter are congruent circles. 

Q.3. What is congruence in nature?
Ans: If the angles of one shape are equal to another, then angles are also congruent. Similarly, if the sides of one shape are equal in length and correspond to another shape, the sides are also congruent in nature.

Q.4. What are congruent objects? Give examples.
Ans: Congruent objects are objects that are identical in nature. They have the same size and shape. They overlap when placed one over the other.
Examples of congruent objects are bangles in a box are congruent, a pair of earrings, etc.

Q.5. What is congruence used for?
Ans: Congruency is used in several ways in our day-to-day life. In any manufacturing industry, they always keep the size of their commodity the same and equal. Consider a milk factory. The packet of \(1\)-litre milk is of equal size and congruent. A biscuit packet will have all the biscuits congruent, which helps the biscuits to fit in the biscuit packet easily.

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