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Lesson 8Triangles with 3 Common Measures

Let’s contrast triangles.

Learning Targets:

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

8.1 3 Sides; 3 Angles

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

Set 1:

6 triangles all have the side lengths of 4, 6, and 9.

Set 2:

4 triangles of different sizes have angles of 41, 56, and 83 degrees.

8.2 2 Sides and 1 Angle

Examine this set of triangles.

9 triangles all have one angle of 30 degrees. All of the triangles also have two side lengths of 5 and 7.
  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.

8.3 2 Angles and 1 Side

Examine this set of triangles.

8 triangles all have two angles of 40 and 80 degrees. All of the triangles also have a side length of 6.
  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.

Are you ready for more?

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Lesson 8 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.

Lesson 8 Practice Problems

  1. Are these two triangles identical? Explain how you know.

    Two triangle are shown both with a 12 unit side length and 70 and 95 degree angles.

  2. Are these triangles identical? Explain your reasoning.

    A quadrilateral where a line segment is drawn from the top vertice to the bottom right vertice creating two triangles that share a common side. Starting from the bottom left angle, going clockwise, the triangle on the left has the angle measurements of 40 degrees, 70 degrees, and 70 degrees. The triangle on the right, starting at the top angle and going clockwise, has the angle measurements of 40 degrees, 70 degrees, and 70 degrees.
  3. Tyler claims that if two triangles each have a side length of 11 units and a side length of 8 units, and also an angle measuring 100^\circ , they must be identical to each other. Do you agree? Explain your reasoning.

  4. The markings on the number line are equally spaced. Label the other markings on the number line.

    A blank number line with 9 evenly spaced tick marks. Starting on the left, the third tick mark is labeled negative 3, the fourth tick mark is labeled 0, and the sixth tick mark is labeled 6.

  5. A passenger on a ship dropped his camera into the ocean. If it is descending at a rate of -4.2 meters per second, how long until it hits the bottom of the ocean, which is at -1,875 meters? 

  6. Apples cost $1.99 per pound.

    1. How much do 3 \frac{1}{4} pounds of apples cost?
    2. How much do x pounds of apples cost?
    3. Clare spent $5.17 on apples. How many pounds of apples did Clare buy?

  7. Diego has a glue stick with a diameter of 0.7 inches. He sets it down 3.5 inches away from the edge of the table, but it rolls onto the floor. How many rotations did the glue stick make before it fell off of the table?