Lesson 17Drawing Triangles

Learning Goal

Let’s see how many different triangles we can draw with certain measurements.

Learning Targets

Given two angle measures and one side length, I can draw different triangles with these measurements or show that these measurements determine one unique triangle or no triangle.
Given two side lengths and one angle measure, I can draw different triangles with these measurements or show that these measurements determine one unique triangle or no triangle.

Warm Up: Using a Compass to Estimate Length

Draw a $40^{\circ}$ angle.
Use a compass to make sure both sides of your angle have a length of 5 centimeters.
If you connect the ends of the sides you drew to make a triangle, is the third side longer or shorter than 5 centimeters? How can you use a compass to explain your answer?

Activity 1: How Many Can You Draw?

Problem 1

Draw as many different triangles as you can with each of these sets of measurements:

open applet in presentation mode

Two angles measure $60^{\circ}$ , and one side measures 4 cm.
Two angles measure $90^{\circ}$ , and one side measures 4 cm.
One angle measures $60^{\circ}$ , one angle measures $90^{\circ}$ , and one side measures 4 cm.

Print Version

Draw as many different triangles as you can with each of these sets of measurements:

Two angles measure $60^{\circ}$ , and one side measures 4 cm.
Two angles measure $90^{\circ}$ , and one side measures 4 cm.
One angle measures $60^{\circ}$ , one angle measures $90^{\circ}$ , and one side measures 4 cm.

Problem 2

Which of these sets of measurements determine one unique triangle? Explain or show your reasoning.

Are you ready for more?

Three equilateral triangles in a line sharing a vertex.

In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a way to move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles.

Activity 2: Revisiting How Many Can You Draw?

Problem 1

Use the applet to draw triangles.

Draw as many different triangles as you can with each of these sets of measurements:

open applet in presentation mode

One angle measures $40^{\circ}$ , one side measures 4 cm and one side measures 5 cm.
open applet in presentation mode
Two sides measure 6 cm and one angle measures $100^{\circ}$ .
open applet in presentation mode

Print Version

Draw as many different triangles as you can with each of these sets of measurements:

One angle measures $40^{\circ}$ , one side measures 4 cm and one side measures 5 cm.
Two sides measure 6 cm and one angle measures $100^{\circ}$ .

Problem 2

Did either of these sets of measurements determine one unique triangle? How do you know?

Lesson Summary

A triangle has six measures: three side lengths and three angle measures.

If we are given three measures, then sometimes, there is no triangle that can be made. For example, there is no triangle with side lengths 1, 2, 5, and there is no triangle with all three angles measuring $150 °$ .

Two drawings illustrating that three side lengths of 1, 2, and 5 cannot make a triangle, nor can 3 angles of 150.

Sometimes, only one triangle can be made. By this we mean that any triangle we make will be the same, having the same six measures. For example, if a triangle can be made with three given side lengths, then the corresponding angles will have the same measures. Another example is shown here: an angle measuring $45 °$ between two side lengths of 6 and 8 units. With this information, one unique triangle can be made.

Given two side lengths and an angle measure, only one unique triangle can be made.

Sometimes, two or more different triangles can be made with three given measures. For example, here are two different triangles that can be made with an angle measuring $45 °$ and side lengths 6 and 8. Notice the angle is not between the given sides.

Two different shapes of triangles shown but both have sides of 6 and 8 and an angle of 45 degrees.

Three pieces of information about a triangle’s side lengths and angle measures may determine no triangles, one unique triangle, or more than one triangle. It depends on the information.