Lesson 9Drawing Triangles (Part 1)

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.

Warm Up: Which One Doesn’t Belong: Triangles

Which one doesn’t belong?

Activity 1: Does Your Triangle Match Theirs?

Three students have each drawn a triangle. For each description of a student’s triangle:

Drag the vertices to create a triangle with the given measurements.
Compare their measurements to the other side lengths and angle measures in your triangle.
Decide whether the triangle you made must be an identical copy of the triangle that the student drew. Explain your reasoning.

open applet in presentation mode

Jada’s triangle has one angle measuring $75^{\circ}$ .
Andre’s triangle has one angle measuring $75^{\circ}$ and one angle measuring $45^{\circ}$ .
Lin’s triangle has one angle measuring $75^{\circ}$ , one angle measuring $45^{\circ}$ , and one side measuring 5 cm.

Print Version

Three students have each drawn a triangle. For each description:

Draw a triangle with the given measurements.
Measure and label the other side lengths and angle measures in your triangle.
Decide whether the triangle you drew must be an identical copy of the triangle that the student drew. Explain your reasoning.

Jada’s triangle has one angle measuring $75^{\circ}$ .
Andre’s triangle has one angle measuring $75^{\circ}$ and one angle measuring $45^{\circ}$ .
Lin’s triangle has one angle measuring $75^{\circ}$ , one angle measuring $45^{\circ}$ , and one side measuring 5 cm.

Activity 2: 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.

Lesson Summary

Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures $120^{\circ}$ and $100^{\circ}$ :

In the figure a horizontal line segment is drawn and labeled 2. On the left end of the line segment, a dashed line is drawn upward and to the left. The angle formed between the dashed line and the horizontal line is labeled 120 degrees. On the right end of the horizontal line, a dashed line is drawn upward and to the right. The angle formed between the dashed line and horizontal line is labeled 100 degrees.

Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles $90^{\circ}$ and $60^{\circ}$ , there is only one way they can meet up and complete to a triangle:

A triangle drawn with a side length of 4 and a 90 and 60 degree angle.

Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and same angle measures.