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Constructing Triangles. Common Core 7.G.2. Vocabulary. Uniquely defined Ambiguously defined Nonexistent. Triangle Inequality Theorem. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. . z. x. y. Practice.

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constructing triangles

Constructing Triangles

Common Core 7.G.2

vocabulary
Vocabulary
  • Uniquely defined
  • Ambiguously defined
  • Nonexistent
triangle inequality theorem
Triangle Inequality Theorem

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

z

x

y

practice
Practice

Can these measures be the sides of a triangle?

  • 7, 5, 4
  • 2, 1, 5
  • 9, 6, 3
  • 7, 8, 4
practice1
Practice

Can these measures be the sides of a triangle?

  • 7, 5, 4 yes
  • 2, 1, 5 no
  • 9, 6, 3 no
  • 7, 8, 4 yes
example 1
Example 1

Using 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 11
Example 1

If 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 12
Example 1

If 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 13
Example 1

If 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.

practice2
Practice

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

  • 2 and 6
  • 9 and 11
  • 10 and 18

(Be sure to look for patterns!)

practice3
Practice

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

  • 2 and 6
  • 9 and 11
  • 10 and 18

What patterns do you see?

how many triangles can be constructed
How many triangles can be constructed?

Remember our “I can” statement:

“I can determine if 1, more than 1, or no triangles can be constructed given 3 side or 3 angle measures.”

slide13

The organizer below should be filled out and glued in your notebook.

Triangles are:

Nonexistent - If three side lengths or angle measures do not make a triangle, you would say that the triangle is nonexistent because a triangle cannot be formed.

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

Ambiguous – If three angle measures do make a triangle, you would say that the triangle is ambiguous because it creates more than 1 triangle.

ambiguous triangles
Ambiguous Triangles
  • Used when 3 angle measures add up to equal 180˚.
  • 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.
  • This proves why more than 1 triangle can be drawn.
unique triangles
Unique Triangles
  • Used when 3 side lengths are given and satisfy the Triangle Inequality Theorem.
  • Look at the triangle below. It has 3 side lengths that will make a triangle. You can flip it, rotate it, or translate it, but there is still ONLY ONE triangle that can be made.
non existent triangles
Non-existent Triangles
  • When angle measures or side lengths DO NOT satisfy our 2 theorems, no triangles can be created.
  • If the sum of the 2 smaller sides is NOT greater than the longest side, it WILL NOT make a triangle.
  • If the sum of the angle measures DO NOT equal 180˚, it WILL NOT make a triangle.
angles of triangles
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.

practice4
Practice

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

Do these measurements create triangles?

3.

4.

practice5
Practice

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

Do these measurements create triangles?

3. yes

4. no

constructing triangles from angles
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 angles1
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 angles2
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 angles3
Constructing Triangles from Angles

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

summary
Summary

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