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The physical world is full of figures with similar shapes. Actually, even the exact same shapes are found! These figures have a few specific mathematical properties that are used in different situations. This lesson introduces the concepts of congruence and similarity and how they are applied in such cases.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Transformation
Translation
Rotation
Reflection
Dilation
Scale Factor
Proportion

Explore

Different Formats of Paper

Paulina loves art class. She paints any chance she gets which is why she has a lot of paper in different formats at home. Take a look at the relationship between two of those sheets of paper.

Two paper sheets where one can be increased or decreased by moving a slider point

What is the relation between the number shown at the top and the quotient of the side lengths of the two sheets of paper?

Discussion

Introducing Rigid Motions

A rigid motion, or isometry, is a transformation that preserves the distance between any two points on the preimage.

A B = A^{'} B^{'}

The following diagram displays two logos. The logo with the points

A

and

B

is the preimage and the logo with the points

A^{'}

and

B^{'}

is the image. The image is the result of a rigid motion because the distances between all points are preserved.

Two logos of Mathleaks with the letters ML and points A, B and their images A' and B'

Rigid motions are also called congruence transformations because the preimage and its image under a rigid motion are congruent figures. Some examples of rigid motions are translations, reflections, and rotations.

When one or more rigid motions are applied to a geometric figure, another figure can be created. Such pair of figures has a specific name.

Discussion

Congruent Figures

Two figures are congruent figures if there is a rigid motion or sequence of rigid motions that maps one of the figures onto the other. As a result, congruent figures have the same size and shape. To denote algebraically that two figures are congruent, the symbol

≅

is used.

Showing that two figures are congruent by mapping one onto the other

When stating that two polygons are congruent, the corresponding vertices must be listed in the same order. For the polygons above, two of the possible congruence statements can be written as follows.

A B C D E ≅ J K L M N or C D E A B ≅ L M N J K

Pop Quiz

Identifying Congruent Figures

Consider different pairs of figures. Are they congruent figures or not?

Different pairs of figures are randomly generated

Discussion

Introducing Similarity Transformation

There is also a type of transformation that creates an image that is not identical, but very similar to its preimage.

Concept

Similarity Transformation

A combination of rigid motions and dilations is called a similarity transformation. The scale factor of a similarity transformation is the product of the scale factors of the dilations.

One triangle is mapped onto the other triangle using rigid motions and dilations

Move the slider to create a similarity transformation by combining rigid motions and dilations.

Figures created by applying a similarity transformation have a specific name.

Discussion

Similar Figures

Two figures are similar figures if there is a composition of similarity transformations that maps one figure onto the other. In other words, two figures are similar if they have the same shape and the ratios of their corresponding linear measures are equal. The symbol

\sim

indicates that two figures are similar.

When writing a similarity statement, the corresponding vertices must be listed in the same order as they appear. The relationship between the two given polygons has multiple similarity statements. Consider two of them.

A B C D \sim J K L M or C D A B \sim L M J K

The same definition applies to three-dimensional shapes.

Now, consider one of the possible similarity statements for the given polyhedrons.

A B C D E F G H \sim J K L M N O P R

Note that for two-dimensional figures, all squares are similar and all circles are similar. Similarly, for

3 D

figures, all cubes are similar and all spheres are similar.

Discussion

Corresponding Parts

Consider two figures. One figure is the image of the other under a transformation. The pairs formed by a part of the preimage — a side, angle, or vertex — and the image of that part are called corresponding parts. For example, in the following applet,

A

and

P

are corresponding vertices.

By clicking on each part of

A B C D,

the corresponding part on

P Q R S

will be highlighted. Note that the figures and their corresponding parts can either be congruent or similar.

Congruent or Similar?	Relationship
Congruent	The corresponding sides and angles of the figures are congruent.
Similar	The corresponding sides of the figures are proportional. The corresponding angles of the figures are also congruent.

Discussion

Properties of Similar Polygons

Two polygons are similar if and only if both of the following two properties hold.

The corresponding angles are congruent.
The corresponding sides are proportional.

Consider a pair of similar polygons. Notice how both of these properties hold for these polygons.

Two similar polygons with congruent angles and proportional sides

Example

Admiring Snowflakes

As Paula walked home from art school, snow began to calmly fall. She became giddy with excitement and hoped to catch snowflakes on her gloves. She noticed two large ones.

External credits: @rawpixel.com, @freepic

Although it is believed that there are no two identical snowflakes, these two snowflakes on Paulina's gloves have congruent shapes. The length of a snowflake is measured from tip to tip.

a The length of the first snowflake is

3

millimeters. What is the length of the second snowflake?

b The angle between two adjacent tips on the first snowflake is

6 0^{\circ} .

What is the measure of the corresponding angle on the second snowflake?

Hint

a Use the definition of congruent figures.

b Congruent figures have the same size and shape.

Solution

a It is a given that the two snowflakes on Paulina's gloves are congruent. Recall that congruent figures have the same size and shape. That fact is proven by the diagram that shows two identical looking snowflakes.

External credits: @rawpixel.com

It is given that the length of the first snowflake is $3$ millimeters. Since the snowflakes are congruent, the length of the second snowflake must also be $3$ millimeters.

External credits: @rawpixel.com

b This time the measure of an angle on the second snowflake must be found. Once again, start by analyzing the appearance of both snowflakes.

External credits: @rawpixel.com

The congruence of the snowflakes indicates that they are identical and have the same lengths and angle measures. That means the corresponding angle on the second snowflake has the same measure of $6 0^{\circ} .$

External credits: @rawpixel.com

Example

Visiting a Gallery

On the weekend, Paulina went to the Mathimartical Gallery, which presents pieces of art that are in some way related to math. One room was dedicated to similar and congruent forms.

Two pictures on the wall, one shows footprints on sand, the other shows different leaves

a The two picture frames showing the footprints and leaves have similar shapes. The width of the larger frame is

5.6

feet and the width of the smaller frame is

4.2

feet. What is the scale factor from the smaller frame to the larger one?

b Some leaves on the picture in the smaller frame also have similar shapes. The length of the left smaller red leaf is

6

inches and the length of a larger red leaf is

10

inches. What is the scale factor from the smaller leaf to the larger leaf?

Hint

a Use the definition of a scale factor.

b Divide the length of one leaf by the length of the other leaf.

Solution

a Begin by recalling the definition of a scale factor.

Scale Factor
A scale factor of two similar figures is the quotient of the measure of one figure and the measure of the other figure.

The width of the larger frame is measured to be $5.6$ feet and the width of the smaller frame is $4.2$ feet.

To find the scale factor from the smaller frame to the larger one, divide

4.2

5.6

and simplify the quotient.

Scale Factor = \frac{4 . 2}{5 . 6} = 0.75

Therefore, the scale factor between the smaller and larger frame is

0.75 .

b The scale factor from the larger leaf to the smaller leaf needs to be found. The smaller leaf is

6

inches long and the larger leaf is

10

inches long.

Calculate the scale factor by dividing

6

inches by

10

inches and simplify.

Scale Factor = \frac{6}{1 0} = 0.6

The scale factor from the smaller leaf to the larger leaf equals

0.6 .

Example

The Room of Games

Paulina goes on to enter The Room of Games! She notices beautifully crafted chess sets and playing cards. Their details are different sizes depending on the piece and card. The purpose of the room is clear to her — similarities and congruence in shapes are being displayed across various games.

Two cards Queen of Diamonds and Four of Diamonds, and two pawns

a The two pawns are similar figures with a scale factor from the smaller to the greater pawn of

1.5 .

If the height of the smaller pawn is

5.8

centimeters, what is the height of the bigger pawn?

b The two cards — the Queen of Diamonds and the Four of Diamonds — have similar shapes with the scale factor from the bigger card to the smaller card of

0.9 .

If the top angle on the diamond on the smaller card is

7 2^{\circ},

what is the top angle on the diamond on the bigger card?

Hint

a Multiply the scale factor by the height of the smaller pawn.

b The angle measures of similar figures are congruent.

Solution

a It is known that the two pawns are similar figures. This means that their dimensions have the same ratio and are related to each other by a scale factor. In this case, it equals

1.5 .

Scale Factor = 1.5

The height of the smaller pawn is

5.8

centimeters. Find the height of the bigger pawn by multiplying

5.8

by the scale factor of

1.5 .

5.8 \cdot 1.5 = 8.7 cm

The height of the bigger pawn is

8.7

centimeters.

b The two cards are said to be similar figures where the Queen of Diamonds is a smaller card and Four of Diamonds is a bigger card. Recall that similar figures have congruent angles. The measure of the top angle of the diamond on the smaller card is

7 2^{\circ} .

Top Angle = 7 2^{\circ}

This means that the measure of the top angle of the diamond on the bigger card is also

7 2^{\circ} .

Pop Quiz

Finding or Using the Scale Factor

Consider two similar figures. Using the given information, find the scale factor rounded to two decimal places or the length of either of the figures rounded to the closest integer.

Applet that randomly generates the images of two objects with the given scale factor or the length of one figure.

Closure

Comparing Ratios of Congruent Triangles

Paulina entered the final room of the gallery. It is dedicated to congruent and similar figures in architecture. There is a model of an old castle with two towers with congruent shapes.

Castle with two towers with congruent shapes

External credits: @tohamina

Notice that the roofs of the towers in front of the castle look like congruent triangles.

Castle with two towers whose rooftops look like congruent triangles from the front

External credits: @tohamina

The height of the rooftop of the tower on the left is

10

feet and its base is

8

feet wide. Calculate the ratio of the height to base by dividing

10

8 .

Ratio of Left Tower Roof = \frac{1 0}{8} = 1.25

Since the towers have congruent shapes, the height of the roof of the right tower is also

10

feet and its base is

8

feet wide. Find the ratio of the height to the base of the right tower's roof as well.

Ratio of Right Tower Roof = \frac{1 0}{8} = 1.25

The calculations show that the ratio of the height to the base is the same for both triangles. This means that ratios of congruent polygons side lengths are equal.

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution