Congruence and Congruence of Plane Figures: Two or more objects are said to be congruent if they are equal to each other in all aspects. For example, a person and his image in the mirror are congruent to each other.

In mathematics, congruent figures are the ones with the same shape and the same size. The process of comparing congruent figures is called congruence. We have different types of plane figures, and in this article, we will discuss the congruence of plane figures such as line segment, angles, triangles, quadrilaterals, etc.

Congruence and Congruence of Plane Figures

Let’s look at a person and their reflection in the mirror and compare them. It is said that objects are congruent when they appear to be identical. In the figures below, the person and his reflection in the mirror are identical in every way. Clearly, there is a congruent between the two figures.

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Two or more objects are said to be congruent if they are equal to each other in all aspects. In mathematics, congruent figures are figures with the same shape and the same size. The word congruence comes from the Latin word “in harmony” or “in agreement.”

The process or relation used in the two congruent figures is known as congruence. The congruence applies to both two-dimensional and three-dimensional figures. But in this article, we shall discuss the congruence of plane figures only.

Different types of congruent figures are available in many situations of our real life. For example, look at the two playing cards as shown in the below figure.

Therefore, congruent figures are the ones with the same shape and the same size. The process or relation used in the two congruent figures is known as congruence.

Superposition of Finding the Congruence of Plane Figures

Two or more plane figures such as line segments, angles, triangles, and other figures are said to be congruent if they have the same shape and the same size. The relation used here is known as the congruence of plane figures.

Look at the below-given figures. Are both of them congruent?

No, because both the images have different shapes and also, they have different sizes.

Let us consider one more example, observe the given figures. Are they are congruent?

Yes, Because they have the same size and the same shape. One image superimposes the other. So, they are said to be congruent. Superimpose means that it covers the other image completely if we place one image over the other image. For example, if we place our hand on the image of the hand, they both superimpose.

In mathematics, superimposition tells the congruence.

Congruence of Plane Figures – Line Segments

A line segment is a basic part of the geometry that connects any two points of the plane figure. The sides of the plane figures are line segments. Two line segments having the same length are said to be congruent. In other words, if two line segments are congruent, then they must have equal length.

In the above figures, two line segments \(\overline {AB} \) and \(\overline {CD} \) Have equal length \(10\) units. Two line segments \(\overline {AB} \) and \(\overline {CD} \) Superimposes on each other. So they are congruent to each other. Hence, the line segments \(\overline {AB} \) and \(\overline {CD} \) are congruent lines.

\(\overline {AB} \cong \overline {CD} \)

Congruence of Plane Figures – Angles

If two rays are starting from the same point, the region or space formed between them is called an angle. Two angles are said to be congruent if both the angles match exactly in all aspects and also, they have equal measurements.

Observe the given angles \(\angle ABC\) and \(\angle PQR\):

Here, two angles \(\angle ABC\) and \(\angle PQR\) measure \({40^{\rm{o}}}\) as shown in the figure, and if we trace one angle over the other, it superimposes. So, the two angles \(\angle ABC\) and \(\angle PQR\) are congruent to each other.

\(\angle ABC \cong \angle PQR\)

Congruence of Plane Figures – Triangles

The closed polygon that is formed with three line segments is called the triangle. We have different types of triangles on the basis of angles or sides of the triangle. We know that triangles have equal shapes, and equal measurements are said to be congruent triangles.

Consider two triangles, \(ABC\) and \(PQR,\) as shown below:

Here, two triangles \(ABC\) and \(PQR,\) are congruent to each other as they superimpose each other. If we place one triangle over the other triangle, they match exactly. Thus, these two triangles have the same shape and the same dimensions. Therefore, these two triangles are said to be congruent.

The congruence of the triangles \(ABC\) and \(PQR,\) are represented by \(ABC \cong PQR.\)
Thus, by placing \(\Delta ABC\) over the other \(\Delta PQR,\) vertex \(A\) falls on the vertex \(P,\,B\) falls on \(Q,\,C\) falls on \(R.\)

The congruence of the triangles \(ABC\) and \(PQR\) is given by
1. Vertices: \(A \leftrightarrow P,\,B \leftrightarrow Q,\,C \leftrightarrow R\)
2. Angles: \(\angle A = \angle P,\,\angle B = \angle Q,\,\angle C = \angle R\)
3. Sides: \(\overline {AB} \leftrightarrow \overline {PQ} ,\,\overline {BC} \leftrightarrow \overline {QR} ,\,\overline {AC} \leftrightarrow \overline {PR} \)

In general, we have four congruency rules to observe the congruence and congruence of triangles that are given below:

1. S.S.S (Side-side-side): Two triangles are said to be congruent if the sides (line segments) are equal (same) to the corresponding sides of another triangle.

2. S.A.S (Side-angle-side): Two triangles are said to be congruent if the two sides (line segments) and the included angle are equal (same) to the corresponding sides and the angle of another triangle.

3. A.S.A (Angle-side-angle): Two triangles are said to be congruent if the two angles and the included side (line segment) are equal (same) to the corresponding two angles and the included side of another triangle.

4. R.H.S (Right Angle-Hypotenuse- side): Two triangles are said to be congruent if the hypotenuse and one side of a triangle are the same or equal to the corresponding hypotenuse and one side of another triangle.

Solved Examples on Congruence and Congruence of Plane Figures

Q.1. Observe the given plane figures and say whether they are congruent or not.

Ans: The plane figures shown in the image are angles.
Given \(\angle PQR = {30^{\rm{o}}}\) and \(\angle ABC = {40^{\rm{o}}}\)
We know that two angles are said to be congruent if both the angles match exactly in all aspects and also, they have equal measurements. If we place one image over the other image, it will not superimpose. And, both the images have different measurements.
So, the given figures are not congruent.

Q.2. Observe the given figures and check whether the two triangles are congruent or not.

Ans: Given \(AB\parallel PQ\) and \(AB = PQ.\)
In triangles \(AOB\) and\(POQ,\)
\(\angle OAB = \angle OQP\) (Alternate interior angles)
\(\angle OBA = \angle OPQ\) (Alternate interior angles)
And, \(AB=PQ\) (Given, which is shown with the two marked lines)
By using the A.S.A congruency rule, \(\Delta AOB \cong \Delta POQ\)
Therefore, the triangles formed are congruent to each other.

Q.3. Keerthi has drawn one plane figure, and she has drawn the reflection of the same image as shown below. Observe the given figures and check for congruency.

Ans: The figures drawn by the Keerthi are

We know that two figures are said to be congruent if they superimpose each other if we place one figure over the other. Reflect the second figure and place it on the first one; it superimposes each other.
Hence, the figures drawn by the Keerthi are congruent figures.

Q.4. Observe the congruence in the image of your foot and its mirror image.
Ans: The image of the foot and its mirror image are shown below:

Reflect the first image and place it on the second image, then two images superimpose each other, and they match exactly. We know that two figures are said to be congruent if they superimpose each other if we place one figure over the other. So, the above two figures are congruent.
Hence, the image of your foot and the mirror image are congruent to each other.

Q.5. The circles shown in the below diagram have the centres A and B, and they have the same radius. Prove that \(\Delta APQ \cong \Delta BPQ.\)

Ans: Given the circles have the same radius. So, the two circles are congruent circles.
In triangles\(APQ,\,BPQ,\)
\(AP = BP\) (Radius of the congruent circles)
\(AQ = BQ\) (Radius of the congruent circles)
\(PQ = PQ\) (Common side)
By using the S.S.S congruence rule, \(APQ \cong \Delta BPQ.\)

Summary

In this article, we have studied the definitions of congruence and congruent figures. We have discussed checking the congruence of plane figures with the help of superimposing method. This article also gives the congruence of plane figures like line segments, angles, triangles etc.

We also studied the four methods of congruence rules such as S.S.S, A.S.A, S.A.S and R.H.S. This article has given the solved examples that help us to understand the concepts of congruence easily.

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FAQs on Congruence and Congruence of Plane Figures

Q.1. What is the congruence of plane figures?
Ans: Two or more plane figures are said to be congruent if they have the same shape and the same size are called congruent plane figures.

Q.2. Which methods are used to check the congruence of plane figures?
Ans: The methods used to check the congruence of plane figures are
1. Superimpose
2. Reflection
3. Translation
4. Rotation

Q.3. Is a mirror image is congruent to the given image?
Ans: Yes. The mirror image is congruent to the given image as both have equal measurements.

Q.4. What are the four ways used in the congruence of triangles?
Ans: The four ways of checking congruence in triangles are
1. S.S.S
2. A.S.A
3. S.A.S
4. R.H.S

Q.5. How do you examine the congruence in the line segments?
Ans: Two or more line segments that have equal length are said to be congruent.