Lesson 6

From Parallelograms to Triangles

Let’s compare parallelograms and triangles.

Problem 1

To decompose a quadrilateral into two identical shapes, Clare drew a dashed line as shown in the diagram.

Trapezoid base 1, 6 units, base 2, 2 units, height 4 units.

  1. She said the that two resulting shapes have the same area. Do you agree? Explain your reasoning.

  2. Did Clare partition the figure into two identical shapes? Explain your reasoning.

Problem 2

Triangle R is a right triangle. Can we use two copies of Triangle R to compose a parallelogram that is not a square?

2 identical triangles labeled R

If so, explain how or sketch a solution. If not, explain why not.

Problem 3

Two copies of this triangle are used to compose a parallelogram. Which parallelogram cannot be a result of the composition? If you get stuck, consider using tracing paper.

A three sided figure in a grid. Vertices located at two units right, 1 unit up; 7 units right, 2 units up; and 1 unit right, 3 units up.
Four parallelograms labeled A, B, C, and D.
A:

A

B:

B

C:

C

D:

D

Problem 4

  1. On the grid, draw at least three different quadrilaterals that can each be decomposed into two identical triangles with a single cut (show the cut line). One or more of the quadrilaterals should have non-right angles.

    Image of a grid.
  2. Identify the type of each quadrilateral.

Problem 5

  1. A parallelogram has a base of 9 units and a corresponding height of \(\frac23\) units. What is its area?

  2. A parallelogram has a base of 9 units and an area of 12 square units. What is the corresponding height for that base?

  3. A parallelogram has an area of 7 square units. If the height that corresponds to a base is \(\frac14\) unit, what is the base?

(From Unit 1, Lesson 5.)

Problem 6

Select all the segments that could represent the height if side \(n\) is the base.

A parallelogram with a bottom side labeled m and a right side labeled n. Dashed lines e, f, j, and k are drawn perpendicular to side m, and dashed lines g and h are drawn perpendicular to side n.
A:

\(e\)

B:

\(f\)

C:

\(g\)

D:

\(h\)

E:

\(m\)

F:

\(n\)

G:

\(j\)

H:

\(k\)

(From Unit 1, Lesson 5.)