# Congruent

Have you ever observed that two copies of a single photograph of the same size are identical?

Similarly, ATM cards issued by the same bank are identical.

Such figures are called congruent figures.

You may have noticed an ice tray in your refrigerator.

The molds inside the tray that is used for making ice are congruent.

Have you struggled to replace a new refill in an older pen?

This could have happened because the new refill is not the same size as the one you want to replace.

Remember that whenever identical objects are to be produced, the concept of congruence is taken into consideration in making the cast.

In this chapter, we will learn about congruent shapes in the concept of congruent, using similar illustrative examples.

Check out the interactive simulation to explore more congruent shapes and do not forget to try your hand at solving a few interesting practice questions at the end of the page.

## What is Congruent?

### Congruent: Definition

The word "congruent" means equal in every aspect or figure in terms of shape and size.

Congruence is the term used to describe the relation of two figures that are congruent.

## What is Congruent Definition in Geometry?

Let us do a small activity.

Draw two circles of the same radius and place one on another.

Do they cover each other completely?

Yes, they do.

Hence, we can say that they are congruent circles.

### Examples

Use the following simulation to explore more congruent shapes.

## How are Triangles Congruent?

Now let's discuss the congruence of two triangles.

Look at $$\Delta ABC$$ and $$\Delta PQR$$ below.

Now, measure the sides and the angles of the triangles.

What do you observe?

Do they measure the same?

If you place $$\Delta ABC$$ on top of $$\Delta PQR$$, would they superimpose?

Yes!

We then say the two triangles are congruent.

We can represent this in a mathematical form using the congruent symbol.

$$\Delta ABC \cong \Delta PQR$$

This means $$A$$ falls on $$P$$, $$B$$ falls on $$Q$$ and $$C$$ falls on $$R$$.

Also, $$AB$$ falls on $$PQ$$, $$BC$$ falls on $$QR$$ and $$AC$$ falls on $$PR$$.

This indicates that the corresponding parts of congruent triangles are equal.

Congruent Parts of $$\mathbf{\Delta ABC}$$ and $$\mathbf{\Delta PQR}$$
Corresponding Vertices

A and P

B and Q

C and R

Corresponding Sides

$$\overline{AB}$$ and $$\overline{PQ}$$

$$\overline{BC}$$ and $$\overline{QR}$$

$$\overline{AC}$$ and $$\overline{PR}$$

Corresponding Angles

$$\angle A$$ and $$\angle P$$

$$\angle B$$ and $$\angle Q$$

$$\angle C$$ and $$\angle R$$

Remember that it is incorrect to write $$\Delta BAC \cong \Delta PQR$$ because $$A$$ corresponds to $$P$$, $$B$$ corresponds to $$Q$$ and $$C$$ corresponds to $$R$$.

Important Notes
1. Two figures are congruent if they have the same shape and size.
2. Two angles are congruent if their measures are exactly the same.

## What are Congruent Shapes?

When one shape is placed over the other and if they superimpose one over the other, they are said to be congruent.

They match exactly even if they are rotated or flipped.

 Example 1

Emma is doing craftwork.

She has different squares of different sizes.

She wants two squares that can be placed exactly one over the other.

Can you help her choose ?

Solution

Squares with the same sides will superimpose on each other.

So, Emma should find two squares whose side lengths are exactly the same.

Such shapes are congruent. In the given list we can see that  green and red squares are of same size.

 $$\therefore$$ green and red squares are congruent to each other.
 Example 2

Help Tim to determine which among the following shapes are congruent?

Solution

Look at figure A and figure B.

If you rotate figure A by $$90^{\circ}$$, then it would perfectly fit onto figure B.

Now, consider figure D and figure H.

Do you know what's the mirror image of figure D?

Figure H is the mirror image of figure D.

So, figure D and figure H would perfectly fit one on the other.

 $$\therefore$$ $$\text{Figure A} \cong \text{Figure B}$$, $$\text{Figure D} \cong \text{Figure H}$$
 Example 3

James took a piece of paper and folded that in half.

He spilled a few drops of different colored inks on the one-half side of the paper.

He pressed the two halves of the paper and opened it.

He obtained a beautiful design.

Do you think that the designs on both halves are congruent?

Solution

Yes, these patterns are congruent.

You can check it by cutting the designs along the edges of the pattern on one half and then putting that on the other half.

You will observe that the design on the other halves completely covers the imprint on the other half.

 Yes, the designs on both halves are congruent.

Challenging Questions
 1. Two friends, Jenny and Jolly each drew a triangle with one side length as 15 inches and two angles as $$40^{\circ}$$ and $$65^{\circ}$$ Jenny says that both triangles are congruent. Do you think she is correct? Give reasons for your answer.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

We hope you enjoyed learning about Congruent with the simulations and practice questions. Now you will be able to easily solve problems on congruent shapes.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. How do you prove a shape is congruent?

Two shapes are congruent if they are exact copies of each other and when superimposed, they cover each other exactly.

## 2. What is another word for congruent?

Congruent shapes are also called coinciding shapes.

## 3. What makes an angle congruent?

Two angles are congruent if their measures are exactly the same.

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