Lesson 16Triangles with 3 Common Measures

Learning Goal

Let’s contrast triangles.

Learning Targets

  • I understand that changing which sides and angles are next to each other can make different triangles.

Warm Up: 3 Sides; 3 Angles

Problem 1

Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different?

Set 1:

Six triangles of different sizes and shapes and side lengths shown.

Set 2:

Four triangles of different sizes and shapes with interior angle measures shown.

Activity 1: 2 Sides and 1 Angle

Problem 1

Examine this set of triangles.

Nine triangles shown of different sizes and shapes with one angle measure and two side lengths indicated.

  1. What is the same about the triangles in the set? What is different?

  2. How many different triangles are there? Explain or show your reasoning.

Activity 2: 2 Angles and 1 Side

Problem 1

Examine this set of triangles.

Eight triangles of different sizes and shapes with two angle measures and one side length indicated.

  1. What is the same about the triangles in the set? What is different?

  2. How many different triangles are there? Explain or show your reasoning.

Lesson Summary

Both of these quadrilaterals have a right angle and side lengths 4 and 5:

Two quadrilaterals each with two given side lengths labeled 4 and 5, and a right angle. On the left, the quadrilaterial is a rectangle with the right angle between adjacent side lengths 4 and 5. On the right, the quadrilateral is a trapezoid with the bottom base labeled 5 and one leg labeled 4. There is a right angle between the bottom base and the leg not labeled.

However, in one case, the right angle is between the two given side lengths; in the other, it is not.

If we create two triangles with three equal measures, but these measures are not next to each other in the same order, that usually means the triangles are different. Here is an example:

Two triangles each with two given side lengths labeled 5 and 6, and an angle labeled 32 degrees. For the triangle on the left, the angle labeled 32 degrees is between the adjacent side lengths 5 and 6. The triangle on the right has the angle labeled 32 degrees between the side length labeled 5 and the third side of the triangle that is not labeled.