How to Find Similar and Congruent Figures?

Congruent figures are those that have sides of the same length and angles. In this article, you learn more about congruence and congruent figures.

How to Find Similar and Congruent Figures?

When one figure overlaps another, the figures are called congruent figures. They fit exactly on top of each other, even when rotated or flipped.

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Step by step guide to finding similar and congruent figures

The word congruent means ‘exactly equal’ in terms of shape and size. Even when we turn, flip, or rotate shapes, they remain the same. For example, draw two circles with the same radius, then cut them and place them on top of each other. We will find that they are on top of each other, that is, they are completely on top of each other. This shows that the two circles are congruent.

The following circles are congruent due to their equal radius and can be placed exactly on top of each other. The symbol that is used to show the congruence of figures is \(≅\). Since circle \(A\) is congruent to circle \(B\), we can state this fact as follows: \(Circle A ≅ Circle B\).

Congruent figures

The congruence of any two figures can be seen if they can be placed exactly over each other. The word “congruence” is used to describe the relationship between two forms that are said to be congruent. In other words, if both geometrical figures can be placed on top of each other, they are called congruent figures. This property applies to all shapes such as triangles, quadrilaterals, and so on. Apart from shapes, line segments and angles are also called congruent if they are of equal size.

See the figure below to understand what congruent numbers mean.

Congruent and similar figures

There is a difference between congruent and similar figures. Congruent figures have the same corresponding side lengths and the corresponding angles are of the same size.  However, similar figures may have the same shape, but their size may not be the same.

For example, look at the following triangles that show the difference between congruent and similar figures. In the congruent figures, we see that all the corresponding sides and angles are of equal measure. However, if we look at similar shapes, we see that the corresponding angles are of equal measure, but the sides are not of equal length.

Congruence of triangles

Two triangles are called congruent if their sides are equal in length, the angles are of equal measure, and they can be superimposed on each other.

In the figure above, \(Δ ABC\) and \(Δ PQR\) are congruent triangles. This means that the corresponding angles and the corresponding sides in both the triangles are equal.

Sides: \(AB = PQ, BC = QR\) and \(AC = PR\);
Angles: \(∠A = ∠P, ∠B = ∠Q\), and \(∠C = ∠R\).
Therefore, \(Δ ABC ≅ Δ PQR\)

The following are the congruence theorems or the triangle congruence criteria that help to prove the congruence of triangles.

  • SSS (Side, Side, Side)
  • SAS (side, angle, side)
  • ASA (angle, side, angle)
  • AAS (angle, angle, side)
  • RHS (Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem)

Finding Similar and Congruent Figures – Example 1:

Are the following triangles similar, congruent, or neither?

Solution:

We know the measures of a corresponding pair of angles and the lengths of a corresponding pair of sides. The sides are opposite the known angle. Therefore, we cannot recognize whether these triangles are similar or congruent.

Exercises for Finding Similar and Congruent Figures

Determine if the triangles are similar, congruent, or neither.

This image has an empty alt attribute; its file name is answers.png
  • \(\color{blue}{Similar}\)
  • \(\color{blue}{Congruent}\)

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