Lesson 6

From Parallelograms to Triangles

Let’s compare parallelograms and triangles.

6.1: Same Parallelograms, Different Bases

Here are two copies of a parallelogram. Each copy has one side labeled as the base \(b\) and a segment drawn for its corresponding height and labeled \(h\)

2 triangles, base b, height h. On right, b is slanted side and height is outside of triangle, perpendicular to the slanted side
  1. The base of the parallelogram on the left is 2.4 centimeters; its corresponding height is 1 centimeter. Find its area in square centimeters.
  2. The height of the parallelogram on the right is 2 centimeters. How long is the base of that parallelogram? Explain your reasoning.

6.2: A Tale of Two Triangles (Part 1)

Two polygons are identical if they match up exactly when placed one on top of the other.

  1. Draw one line to decompose each polygon into two identical triangles, if possible. If you choose to, you can also draw the triangles.
  2. Which quadrilaterals can be decomposed into two identical triangles?

    Pause here for a small-group discussion.

  3. Study the quadrilaterals that were, in fact, decomposable into two identical triangles. What do you notice about them? Write a couple of observations about what these quadrilaterals have in common.



Draw some other types of quadrilaterals that are not already shown. Try to decompose them into two identical triangles. Can you do it? Come up with a general rule about what must be true if a quadrilateral can be decomposed into two identical triangles.

GeoGebra Applet XpEZQfqp

6.3: A Tale of Two Triangles (Part 2)

This applet has eight pairs of triangles. Each group member should choose 1–2 pairs of triangles. Use them to help you answer the following questions.

    1. Which pair(s) of triangles do you have?

    2. Can each pair be composed into a rectangle? A parallelogram?

  1. Discuss your responses to the first question with your group. Then, complete each of the following statements with all, some, or none. Sketch 1–2 examples to illustrate each completed statement.
    1. ________________ of these pairs of identical triangles can be composed into a rectangle.

    2. ________________ of these pairs of identical triangles can be composed into a parallelogram.

Summary

A parallelogram can always be decomposed into two identical triangles by a segment that connects opposite vertices.

Three parallelograms showing decompositions into two identical triangles.

Going the other way around, two identical copies of a triangle can always be arranged to form a parallelogram, regardless of the type of triangle being used.

To produce a parallelogram, we can join a triangle and its copy along any of the three sides, so the same pair of triangles can make different parallelograms.

Here are examples of how two copies of both Triangle A and Triangle F can be composed into three different parallelograms.

Six parallelograms composed from two identical triangles.

This special relationship between triangles and parallelograms can help us reason about the area of any triangle.

Glossary Entries

  • base (of a parallelogram or triangle)

    We can choose any side of a parallelogram or triangle to be the shape’s base. Sometimes we use the word base to refer to the length of this side.

    Three identical, scalene, obtuse triangles oriented the same way. Each triangle has a different one of its sides labeled base.
  • height (of a parallelogram or triangle)

    The height is the shortest distance from the base of the shape to the opposite side (for a parallelogram) or opposite vertex (for a triangle).

    We can show the height in more than one place, but it will always be perpendicular to the chosen base.

    2 parallelograms with height measurements
  • parallelogram

    A parallelogram is a type of quadrilateral that has two pairs of parallel sides.

    Here are two examples of parallelograms.

    Two paralllelograms with the angles and side lengths provided.
  • quadrilateral

    A quadrilateral is a type of polygon that has 4 sides. A rectangle is an example of a quadrilateral. A pentagon is not a quadrilateral, because it has 5 sides.