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Congruent Triangles (and other shapes)

Home > By Subject > Geometry > Congruent Triangles

Working through the lesson below will help your child to understand that congruent figures can be determined by a figure rotation, reflection, or translation or any combination of the three. It will also help them to identify the types of transformations in a sequence.

Learning Takeaways: After this lesson, students will be able to:

  • Identify the characteristics of congruent figures
  • Perform transformations on a figure (rotation, reflection, and translation)
  • Identify the types of transformations in a sequence

Make sure your child is familiar with the vocabulary below:

  • Two-dimensional figure: A figure that lies in a plane.

Congruent Triangles (and other figures)

This section will help your child to identify the characteristics of congruent figures.

Two figures are congruent if they are the:

  • Exact same shape
  • Exact same size
    • Angle measures are equal
    • Line segments are equal

Look at the example below.

congruent triangles after translation

These triangles are congruent. They are the exact same size AND shape. If you slid triangle A to the right, it would exactly cover triangle B. This is called a translation. You will learn more about translations in the next section of this lesson.

Discuss the examples and questions below with your child regarding whether the figures are congruent.

two non-congruent rectangles These rectangles are not congruent. They are not the same size.
two triangles that are not congruent These triangles are not congruent. They are the same size but not the same shape. Triangle B is a right triangle. Triangle A is an isosceles triangle.
two congruent parallelograms

Are these two parallelograms congruent? Are they the exact same shape and the exact same size?

Answer: They are the same shape and size so they are congruent. See more about rotations later in this lesson.

Which figure is congruent to figure C shown below?
5 different shapes two of which are congruent

Figure b. is congruent.

Transformations : Rotations, Reflections, & Translations

This section will help your child to perform a transformation (rotation, reflection, and translation) on a figure .

Make sure your child is familiar with the vocabulary below:

  • Transformation moves a figure from its original place to a new place.
  • Angle of Rotation: How big the angle is that you rotate a figure.  Common angle rotations are 45°, 90°, 180°.
  • Isometric Transformation: A transformation that does not change the size of a figure.

There are three types of transformations. Alternative names are in parenthesis:

  1. Rotation (Turn): Turns a figure around a fixed point. 
  2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
  3. Translation (Slide or glide): Sliding a shape to a new place without changing the figure.

Rotations, reflections, and translations are isometric.  That means that these transformations do not change the size of the figure.  If the size and shape of the figure is not changed, then the figures are congruent.

Explore and discuss the examples of transformations below with your child.

Examples of Transformations

Rotation
congruent triangles via rotation

Triangle A is a 90° rotation of triangle B.

The angle of rotation of is 90 degrees. Notice how the angle created between the 2 figures is equal to the angle of rotation.

Reflection
example of reflection transformation

Reflections “flip” a figure over a line (often referred to as a line of symmetry).

Reflections are mirror images and appear “backwards” from the original figure.

Translation
example of translation transformation

A translation slides or glides a figure from one place to another.  A translation cannot have any rotation (or else it would be a rotation).

Try It! Find a flat object in your home that can easily be moved (small book, calculator, drink coaster, coin, etc.)  Perform each transformation using that object.

Multiple Transformations

This section will help your child to understand that congruent figures can have more than one transformation.

Make sure your child is familiar with the vocabulary below:

  • Sequence: A group of things arranged in a certain order.  Commonly known as a pattern.

Recapping from earlier in his lesson, there are three types of transformation:

  1. Rotation (Turn): Turns a figure around a fixed point. 
  2. Reflection (Flip): Flip of figure over a line where a mirror image is created.
  3. Translation (Slide or glide): Sliding a shape to a new place without changing the figure.
Two Transformations

example of transformation

Triangle B has performed two transformations.  It is rotated 90° and translated.

Triangle B is a rotation of triangle A because it turned 90°.  It was also slid up and to the right, making a translation.

example of transformation

 

What two transformations could have been performed here? Hint: Figure 2 is a mirror image of figure 1.

Figure 2 is a reflection and translation of figure 1. The figure is a reflection because it flipped. It is a translation because it is moved to another place, without rotating it. This is also called a glide reflection.

Try It! Look at the figure below. What transformations does parallelogram Z perform?

example of 2 transformations

Think: Does figure Z face the same direction? Are corresponding parts of the parallelogram parallel each other? Are the figures in the same quadrant of the Cartesian plane?

Congruent Figures Worksheets

Click the links below and get your child to try the worksheets on congruent figures and practice with questions based on what is shown above.

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