Warm Up/ Activator

Using scissors and a ruler, cut out a triangle from a piece of colored paper. Clean up your area afterwards. We will be using the triangle during our lesson today.

Constructing Triangles

Common Core 7.G.2

Vocabulary• Uniquely defined• Ambiguously defined• Nonexistent

Triangle Inequality Theorem

Let’s review what we learned yesterday with this video.

http://www.youtube.com/watch?v=OoLb_NnnKSQ

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

x z

y

PracticeCan these measures be the sides of a triangle?

1. 7, 5, 42. 2, 1, 53. 9, 6, 34. 7, 8, 4

PracticeCan these measures be the sides of a triangle?

1. 7, 5, 4 yes2. 2, 1, 5 no3. 9, 6, 3 no4. 7, 8, 4 yes

Example 1Using the measurements 6 in and 8 in, what is the smallest possible length of the third side? What is the largest possible length of the third side?

Example 1If you assume that 6 and 8 are the shorter sides, then their sum is greater than the third side. Therefore, the third side has to be less than 14.

Example 1If you assume that the larger of these values, 8, is the largest side of the triangle, then 6 plus the missing value must be greater than 8. Therefore, the third side has to be more than 2.

Example 1If you put these two inequalities together, then you get the range of values that can be the length of the third side:

Therefore, any value between 2 and 14 (but not equal to 2 or 14) can be the length of the third side.

PracticeSolve for the range of values that could be the length of the third side for triangles with these 2 sides:

1. 2 and 62. 9 and 113. 10 and 18

(Be sure to look for patterns!)

PracticeSolve for the range of values that could be the length of the third side for triangles with these 2 sides:

1. 2 and 6 2. 9 and 113. 10 and 18

What patterns do you see?

Types of TrianglesIf three side lengths do not make a triangle, you would say that the triangle is nonexistent.

If three side lengths do make a triangle, you would say that the triangle is uniquely defined because it creates one, specific triangle.

Angles of TrianglesTear off the corners of the triangle that you created in the warm up/activator. Lay them on your paper with all of the vertices pointing inwards and the edges touching. Like this:

What do they create?

Angles of Triangles

What do they create? A straight line, which is equal to 180 degrees; therefore, the sum of the angles in a triangle always equal 180 degrees. This is called the Triangle Angle Sum Theorem. Glue your triangle corners in your math notebook and explain this in your own words.

PracticeGiven the following angle measurements, determine the third angle measurement.

Do these measurements create triangles?3. 4.

PracticeGiven the following angle measurements, determine the third angle measurement. 1.2.

Do these measurements create triangles?3. yes4. no

Constructing Triangles from Angles

Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths?

Constructing Triangles from Angles

Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths? No.

Since they aren’t the same size, will angle measurements construct unique triangles?

Constructing Triangles from Angles

Look at these triangles. They have the same angle measurements, which is why they are similar in shape. However, do they have the same side lengths? No.

Since they aren’t the same size, will angle measurements construct unique triangles? No.

Constructing Triangles from Angles

Conditions, such as angle measurements, that can create more than one triangle are called ambiguously defined.

SummaryTake turns with your partner explaining the Triangle Angle Sum Theorem and the Triangle Inequality Theorem in your own words.