Topics
8. Transformations & Congruence
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United States of America
Grade 8

8.04 Congruence and transformations

Lesson
Practice

Introduction

We have already learned that a transformation is a process that manipulates a polygon or other two-dimensional objects on a plane or coordinate system. A transformation that doesn't change the size or shape of a geometric figure is called rigid transformation.

These transformations are known as:

  • translations

  • reflections

  • rotations

We will now determine congruence by examining two figures and identifying the rigid transformation(s) that produced the figures.

Congruence and transformations

We say two shapes X and Y are congruent if we can use some combination of translations, reflections, and rotations to transform one shape into the other. For example, the following pair of triangles are congruent:

A pair of congruent triangles.

We use the symbol \cong to express congruence, so we read X \cong Y as "X is congruent to Y".

Examples

Example 1

Consider \triangle ABC and \triangle A''B''C''

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Are the two triangles congruent? How do you know?

Worked Solution
Create a strategy

We can say that the two shapes are congruent if we can use some combination of reflections, translations, and rotations to align the two shapes exactly.

Apply the idea

Find some combination of transformations to align the two shapes.

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  • If we reflect \triangle ABC across the y-axis, then we'll have\triangle A'B'C'.

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  • We can then translate \triangle ABC,\, 10 units to the left and 9 units down, so that we'll have \triangle A''B''C''.

We can now say that \triangle ABC \cong \triangle A''B''C''

Reflect and check

Now when looking at the original shapes we also know which sides and angles correspond to each other.

Example 2

State the two types of transformations that would be needed to get from Flag A to Flag B.

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Worked Solution
Create a strategy

Let's experiment with the various transformations and attempt to find the possible combinations until we find the pair that works.

Apply the idea

Ask the following questions:

What happens to Flag A if you reflect it? Rotate it? Translate it?

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  • Rotating Flag A by 90\degree with respect to the origin will result in an orientation similar to Flag B.

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  • Translating the image 1 unit to the left and 3 units up will result in Flag B.

The two types of transformations to get from Flag A to Flag B are rotation and translation.

Idea summary

Two shapes X and Y are congruent if we can use some combination of rigid transformations to transform one into the other.

We use the symbol \cong to express this relationship, so we read X\cong Y as "X is congruent to Y".

Outcomes

8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

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