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Congruent Shapes

Introduction

Before learning about congruent shapes, we have to learn first about the meaning of congruent in Mathematical Geometry. The word congruent describes those shapes and figures that can be transferred or twisted to match with the other shapes. Two shapes are said to be congruent shapes if they have a similar shape and similar size. In other words, we can say about the congruent that if the mirror image of one shape is the same as the other shape. The concept of congruence in triangles will be defined as if two triangles are said to be congruent if all three corresponding sides of a triangle are equal and also all three corresponding angles are equal in measure. These congruent triangles can be rotated, slides, twisted, and turned to seem the same. If rearranged, they match with each other, having the same shape and size and equal angles. The corresponding side of the triangle and angles of the triangle for congruent triangles will be equal to each other. We have four congruence rules to prove that two triangles are congruent. But it is not necessary to find all six values to prove the triangle is congruent. We can find the congruence of the triangle by finding only three values either sides or angles. The actual meaning of congruent in math is when two shapes are similar to each other based on their shapes،size and angles in the case of the triangle. Congruence is the term that is used to explain an object and its mirror image as a whole. Two shapes or objects are called congruent if they coincide with each other. Their shapes and all elements are the same. In the case of geometric shapes, position vectors with the same length or radius are congruent, and angles with the same estimate are congruent. Let’s learn about congruent shapes in detail below

Definition

Two Shapes mathematically will be considered to be congruent shapes :

If they have identical figures and sizes with equal corresponding angles congruent, and the length of corresponding sides are also congruent.

OR

Two shapes with the same shape and sizes are called congruent shapes.

Congruent shapes have the same shape and size. They will remain congruent after translation, reflection and rotation.

Transformations

Congruent Shapes include the following transformation:

Translation,

Shapes: the same

Sizes: the same will be Congruent Shapes

Reflection,

Shapes: the same

Sizes: the same will be Congruent Shapes

Rotation,

Shapes: the same

Sizes: the same will be Congruent Shapes

Extension or reduction,

Shapes: the same

Sizes: the same will be Congruent Shapes

If from shape or size one is different then the shape will not be the congruent shape.

When two triangles are congruent then their corresponding angles are congruent and the lengths of corresponding sides are also the same. These triangles are called Congruent triangles.

Similarly, when two polygons are Congruent then their corresponding angles are congruent and the lengths of corresponding sides are also the same. These polygons are called Congruent polygons.

We can generalize this definition of Congruent Shapes for shapes that fulfill the definition of Congruent Shapes then these shapes will be congruent.

Which things make two shapes “congruent”?

Following things make a pair of shapes congruent if they have

  1. Same Corresponding Angles
  2. Same Corresponding side lengths
  3. Same size
  4. Can be rotated or a mirror image
  5. A cut-out of one shape will always fit exactly over the other shape
  6. Three Sides Equal in terms of congruent triangles
  7. Two Sides and Their Included angle equal in terms of congruent triangles
  8. Two Angles and One Side Equal in terms of congruent triangles
  9. Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides in terms of congruent triangles.

Properties of Congruent Shapes

The properties of congruence apply to lines, angles, and figures. They can be described as follows:

  1. Reflexive property
  2. Symmetric property
  3. Transitive property

Now we will explain these properties one by one.

The reflexive property of congruence says that a line segment, an angle, or a shape is always congruent to itself. For example, ∠A≅∠A

Now for a second property

The symmetric property says that if one shape is congruent to another shape, then the second shape is also congruent to the first shape. For any two angles A and B if ∠A≅∠B, then ∠B ≅∠A.

The transitive property of congruence states that if the line of the first shape is congruent to the line of the second shape, and the line of the second shape is congruent to the line of the third shape, then a line of the first shape is also congruent to the line of the third shape.

Simple Congruence

For most shapes when we say that these are congruent then it is as simple as determining if they are the same size and same shape

Example

Congruency is what two shapes have if they’re congruent.

One rectangle points are up, while the other points are towards the right. However, when we will measure these two rectangles, we found that the corresponding sides of both rectangles are the same. The second triangle is the same as the first; it’s just been rotated from up to right. So, these shapes are congruent. It’s important to remember that the directions shapes face don’t affect the congruency of shapes. We have seen that both rectangles have the same shape and have the exact measurements, they’re congruent

Similarly, we can see any two solid circles and check their congruence

Both shapes A and B show congruency because they’re the same size and shape as each other.

In both examples, we have seen the relation of congruency between two shapes and conclude that congruence shapes have the same size and shapes.

Congruence in Triangles

Congruence in two or more triangles in studying congruent triangles will depend on the estimation of triangle sides and angles of triangles. Basically in congruent triangles, we know that the three sides of a triangle find their size, and the three angles of a triangle find their shape. Simply, two triangles are said to be congruent if their corresponding sides and their corresponding angles are equal to each other. They have the same shape and same size. Let us discuss congruent triangles examples in detail.

Example :

Below are two triangles named Δ ABC and Δ XYZ. Then show that they are congruent triangles?

Solution :

In the figure given above, Δ ABC and Δ XYZ are congruent triangles. This means that the corresponding angles of triangles and corresponding sides of triangles in both the triangles are equal.

Corresponding Sides: AB = XY, BC = YZ and AC = XZ;

Corresponding Angles: ∠A = ∠X, ∠B = ∠Y, and ∠C = ∠Z.

We found that corresponding sides and angles are equal.

Therefore, Δ ABC ≅ Δ XYZ.

Some Theorems for the Congruence of Triangles

  • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
  • If two triangles have all three sides in common then they are congruent.
  • When two triangles have two angles that are the same, and the side length between them is also the same, they are congruent.
  • Two triangles with two sides the same and an angle in between them the same are congruent.
  • Two triangles with a right-angle, the hypotenuse, and the adjacent length the same will be congruent.

Area of Congruent Shapes

We know that every congruent shape is also a similar shape but by the concept of area of similar shape is also similar we can conclude that the area of congruent shape will be congruent.

For calculating an unknown area in the congruent shapes, first, we need to calculate the Area of the first shape then it will be the same area for another shape because in the congruent shape area is also congruent but if the area is congruent we can not say that shapes are congruent.

In other words, the area of a congruent shape is congruent but the inverse is not true. Let’s check this fact through examples.

For example,

We have two circles A and Circle B with radius r = 8 and these two circles are congruent to each other then find that they have a congruent area?

Solution :

Given that radius of circle A is 8 cm,

We know that by area of the circle

Area of circle:  π × r2   (where r is the radius of the circle)

Now putting the value of r in the above formula

Area of circle =  π × 82

We know that the value of π  is 3.1415 or 227            

                                                    =  3.1415 × 8 ×8

                                                    =   201.056 cm2

For the second circle area, we know that the radius size is the same, so the area of circle B will be similar to circle A.

Hence, the area of circle A and circle B is congruent.

How to measure

We can measure the area of congruent shapes by finding the area of the first shape then the area of the second shape will be congruent to the first shape. We know that the following formulas are used to find the areas of different shapes

Area of Triangle:  $\frac{1}{2}$ × b  × h   ( where b is base and h is the height of the triangle )

Area of Rectangle:  l × w    ( where I is length and w is the width of rectangle )

Area of parallelogram: b  × h  ( Where b is base and

h is vertical height parallelogram )

Area of circle:  π × r2  (where r is the radius of the circle)

Area of the ellipse:  π × a × b    ( where a is $\frac{1}{2}$  minor axis and b is $\frac{1}{2}$ major axis )

Area of square:  a2   ( where a is the length of circle side )

We will find some congruent shapes areas to clear the concept of congruent area in different congruent shapes.

For Example,

We  have  two parallelogram ABCD and parallelogram EFGH  with parallel side a = 8, b =4 and for second parallelogram x =8 , y= 4 also with same height h = 5. The angle between any two sides of a parallelogram is a right angle  and these two parallelograms are congruent to each other then

Find the area of these parallelograms is congruent?

Solution :

For parallelogram ABCD,

We have a= 8 and b= 4, then by parallelogram area formula

Area of parallelogram = b  × h 

Area of parallelogram = 4  ×  5 

Area of parallelogram = 20  sq.unit

Now for  parallelogram EFGH,

We have x= 8 and y= 4, then by parallelogram area formula

Area of parallelogram = y  × h 

Now putting the value of y and h in the above formula

Area of parallelogram = 4  ×  5 

Area of parallelogram = 20  sq.unit

Note that area of both parallelograms is congruent to each other.

Finding area without the height of parallelogram?

We see this fact  for  parallelogram EFGH,

We have x= 8 and y= 4, then by parallelogram area formula  without height,

Area of parallelogram = x × y  sin θ

Where θ is the angle between any two sides of the parallelogram which is right angle so, θ  = 90°

Now putting the value of x , y and θ in above formula

               Area of parallelogram = 8 × 4 sin 90°                  sin 90° = 1

                Area of parallelogram = 8 × 4 × 1

                 Area of parallelogram = 32 sq.unit

Similarly, we can see that for parallelogram ABCD, the angle between any two sides of the parallelogram which is the right angle, the area will be the same as parallelogram EFGH.

Therefore, these two parallelogram ABCD and parallelogram EFGH  have congruent areas.

Volume of Congruent Shapes

We know that every congruent shape is also similar shape than by the concept of volume of similar shape is also similar we can conclude that the volume of congruent shape will be congruent.

For calculating an unknown volume in the congruent shapes, first, we need to calculate the volume of the first shape then it will be the same volume for another shape because in the congruent shape volume is also congruent but if the volume is congruent we can not say that shapes are congruent.

In other words, the volume of congruent shape is congruent but the inverse is not true. Let’s check this fact through examples

For example,

We have two cones with radius r, slant height l, and h height. If r= 3 cm, l =5 cm and h =4 cm for both cones then find the volume of these cones and show that volume is congruent?

Solution :

We know that from both shapes these cones are equal in shape and size so, they are congruent to each other.

For volume, we have the formula

Volume of cone = 13  πr²h

Now putting the value of r, π, and h in the above formula

Volume of cone = 13  × 3.1415 × 3² × 4

Volume of cone = 37.698 cm3

Hence the volume for both cones will be similar because both shapes are congruent, so their volume will be congruent.

How to measure

We can measure the volume of congruent shapes by finding the volume of the first shape then the volume of the second shape will be congruent to the first shape. We know that the following formulas are used to find the areas of different shapes

Rectangular Solid or Cuboid:  l × w × h     (Where l is Length w  is Width and h = Height )

Cube: a³    ( Where a is Length of edge or side )

Cylinder:  πr²h    ( Where r is Radius of base and h is Height )

Sphere:  43 πr³     ( Where r is Radius of the sphere )

We will find some congruent shapes areas to clear the concept of congruent area in different congruent shapes.            

Solved Examples

Example 1 :

Given below are two right-triangles with  the  same hypotenuse and the adjacent length

Show thatΔ ABC is congruent to  Δ LMN?

Solution :

We know that two sides in  Δ ABC are equal to the corresponding hypotenuse and the adjacent length of ΔLMN  with one right angle equal to each other.

                                         BC   ≅ LM because   8 cm = 8 cm

Also, the angle of A is equal to the angle of L,

                                       ∠L  ≅ ∠P because   ∠ 90° = ∠90°      

We know that right-angle = 90°      

Hence, we have seen that one right-angle of both triangles is equal, hypotenuse and the adjacent length are the same so,  Δ ABC  is congruent to   Δ LMN.

Example 2

Given below are two triangles with two equal sides and have equal angles between these sides.

Show that  Δ LMN is congruent to  Δ PQR?

Solution :

We know that two sides in  Δ LMN  are equal to the corresponding two sides of ΔPQR  with one side angle equal to each other.

                                         LM  ≅ PQ because   9 cm = 9 cm

                                         LN  ≅ PR because   9 cm = 9 cm

Also, the angle of L is equal to the angle of P,

                                       ∠L  ≅ ∠P because   ∠ 60° = ∠60°                                

Hence, we have seen that one angle of both triangles is equal and the corresponding two sides lengths are the same so,  ΔLMN  is congruent to   Δ PQR.

Example 3 :

Given below are two triangles with two equal angles  and have one  equal  side between them

Show that ΔABC is congruent to ΔPQR?

Solution  :

We know that  two angles in Δ ABC  are equal to corresponding two angles of ΔPQR with one side length equal to each other

                                        ∠C  ≅ ∠R because   ∠ 40° = ∠40°

                                        ∠B  ≅ ∠Q because   ∠ 60° = ∠60°

Also, the side length of AB is equal to the side length of PQ,

                                       AB  ≅ PQ because   4 cm = 4 cm

Hence, we have seen that one side length of both triangles is equal and the corresponding two angles are the same so,  ΔABC is congruent to ΔPQR.

Example 4 :

Given below are two triangles with all three equal sides

Show that Δ ABC is congruent to Δ PQR?

Solution  :

We know that all three corresponding sides of ΔABC are equal to corresponding sides of ΔPQR, so we have

 AB  ≅ PQ because   8 cm = 8 cm

BC  ≅ QR because  9 cm = 8 cm

AC  ≅ PR because   3 cm = 3 cm

We have seen that all three sides of both triangles are equal so, they are congruent.

Example 5 :

Show that  Δ ABC is congruent to Δ XYZ?

Solution :

From above these two triangles,  Δ ABC  are congruent to two angles of  Δ XYZ then it is clear from both shapes of triangles that are congruent, we have from the above theorem

                                                 ∠C ≅∠Z

The third angles of  Δ ABC and Δ XYZ are congruent.

Key Facts and Summary

  • Two shapes with the same shape and sizes are called congruent shapes.
  • Congruent shapes have the same shape and size. They will remain congruent after translation, reflection and rotation.
  • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
  • If two triangles have all three sides in common then they are congruent.    
  • When two triangles have two angles that are the same, and the side length between them is also the same, they are congruent.
  • Two triangles with two sides the same and an angle in between them the same are congruent.
  • Two triangles with a right-angle, the hypotenuse, and the adjacent length the same will be congruent.
  • Area and volume of congruent shape are congruent but the inverse is not true.

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