Unit 1, Lesson 7
From Parallelograms to Triangles
Grade 6
7.1
Warm-up
Here are two copies of a parallelogram. Each copy has one side labeled as the base and a segment drawn for its corresponding height and labeled .
Two triangles, base b, height h. On right, b is slanted side and height is outside of triangle, perpendicular to the slanted side.
- The base of the parallelogram on the left is 2.4 centimeters; its corresponding height is 1 centimeter. Find its area in square centimeters.
- The height of the parallelogram on the right is 2 centimeters. How long is the base of that parallelogram? Explain your reasoning.
7.2
Activity
Two shapes are identical if they match up exactly when placed one on top of the other.
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Draw one line to decompose each shape into two identical triangles, if possible. Use a straightedge to draw your line.
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Which quadrilaterals can be decomposed into two identical triangles?
Pause here for a small-group discussion.
- Study the quadrilaterals that can, in fact, be decomposed into two identical triangles. What do you notice about them? Write a couple of observations about what these quadrilaterals have in common.
Student Lesson Summary
A parallelogram can always be decomposed into two identical triangles by a segment that connects opposite vertices.
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 that match, 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.
This special relationship between triangles and parallelograms can help us reason about the area of any triangle.
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