Lesson 9

Composing Figures

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

9.1: Angles of an Isosceles Triangle

Here is a triangle.

  1. Reflect triangle \(ABC\) over line \(AB\). Label the image of \(C\) as \(C’\).
  2. Rotate triangle \(ABC’\) around \(A\) so that \(C’\) matches up with \(B\).

  3. What can you say about the measures of angles \(B\) and \(C\)?

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

9.2: Triangle Plus One

Here is triangle \(ABC\).

  1. Draw midpoint \(M\) of side \(AC\).

  2. Rotate triangle \(ABC\) 180 degrees using center \(M\) to form triangle \(CDA\). Draw and label this triangle.

  3. What kind of quadrilateral is \(ABCD\)? Explain how you know.
Triangle A B C is scalene with side B C as base and longest side, and side AB as the shortest side.


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?

9.3: Triangle Plus Two

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

Rotations of triangle ABC.

  1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points in Triangle 2 correspond to points \(A\), \(B\), and \(C\) in the original triangle?

  2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points in Triangle 3 correspond to points \(A\), \(B\), and \(C\) 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.

9.4: Triangle ONE Plus

Here is isosceles triangle \(ONE\). Its sides \(ON\) and \(OE\) have equal lengths. Angle \(O\) is 30 degrees. The length of \(ON\) is 5 units.

Triangle O N E. Angle N O E is labeled 30 degrees. Side O N is labeled 5.

  1. Reflect triangle \(ONE\) across segment \(ON\). Label the new vertex \(M\).

  2. What is the measure of angle \(MON\)?

  3. What is the measure of angle \(MOE\)?

  4. Reflect triangle \(MON\) across segment \(OM\). Label the point that corresponds to \(N\) as \(T\).

  5. How long is \(\overline{OT}\)? How do you know?

  6. What is the measure of angle \(TOE\)?

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

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 \(ABC\).

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

We can reflect triangle \(ABC\) across side \(AC\) 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 \(A\) and \(C\) are on the line of reflection, they do not move. So the image of triangle \(ABC\) is \(AB'C\). We also know that:

  • Angle \(B'AC\) measures \(36^\circ\) because it is the image of angle \(BAC\).
  • Segment \(AB'\) has the same length as segment \(AB\).

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.

Glossary Entries

  • corresponding

    When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

    For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).

    2 triangles with corresponding parts
  • rigid transformation

    A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations, as is any sequence of these.

  • vertical angles

    Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.

    For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

    Angles \(AED\) and \(BEC\) are another pair of vertical angles.

    a pair of intersecting lines that create vertical angles