Lesson 9Composing Figures

Learning Goal

Let’s use reasoning about rigid transformations to find measurements without measuring.

Learning Targets

  • I can find missing side lengths or angle measures using properties of rigid transformations.

Lesson Terms

  • corresponding
  • rigid transformation
  • vertical angles

Warm Up: Angles of an Isosceles Triangle

Problem 1

    • Reflect triangle over line . Label the image of as .

    • Rotate triangle around so that matches up with .

    A triangle labeled A B C with horizontal side B C labeled 2 and sides A B and A C are each labeled 3.

  2. What can you say about the measures of angles and ?

Activity 1: Triangle Plus One

Problem 1

Here is triangle .

  1. Draw midpoint of side .

    Rotate triangle degrees using center to form triangle . Draw and label this triangle.

    Blue triangle ABC

  2. What kind of quadrilateral is ? Explain how you know.

Are you ready for more?

Problem 1

In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. This picture helps you justify a well-known formula for the area of a triangle. What is the formula and how does the figure help justify it?

Activity 2: Triangle Plus Two

Problem 1

The picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations.

Blue triangle ABC labeled 1. Black triangle ACD labeled 2 and black triangle ABE labeled 3

  1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points in Triangle 2 correspond to points , , and in the original triangle?

  2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points in Triangle 3 correspond to points , , and in the original triangle?

  3. Find two pairs of line segments in the diagram that are the same length, and explain how you know they are the same length.

  4. Find two pairs of angles in the diagram that have the same measure, and explain how you know they have the same measure.

Activity 3: Triangle ONE Plus

Problem 1

Here is isosceles triangle . Its sides and have equal lengths. Angle is 30 degrees. The length of is 5 units.

  1. Reflect triangle across segment . Label the new vertex .

    Triangle ONE. Side ON is 5 units. Angle O is 30 degrees.

  2. What is the measure of angle ?

  3. What is the measure of angle ?

  4. Reflect triangle across segment . Label the point that corresponds to as .

  5. How long is ? How do you know?

  6. What is the measure of angle ?

  7. If you continue to reflect each new triangle this way to make a pattern, what will the pattern look like?

Lesson Summary

Earlier, we learned that if we apply a sequence of rigid transformations to a figure, then corresponding sides have equal length and corresponding angles have equal measure. These facts let us figure out things without having to measure them!

For example, here is triangle .

A triangle A, B, C where the interior angle at A has measure 36 degrees.

We can reflect triangle across side to form a new triangle:

Triangle A, B, C, with angle with measure 36 degrees at A. It has been reflected on the side A, C.

Because points and are on the line of reflection, they do not move. So the image of triangle is . We also know that:

  • Angle measures because it is the image of angle .

  • Segment has the same length as segment .

When we construct figures using copies of a figure made with rigid transformations, we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles.