1. Constructing Triangles
Common Core 7.G.2

2. Vocabulary
Uniquely defined
Ambiguously defined
Nonexistent

3. 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+𝑦>𝑧 𝑥+𝑧>𝑦 𝑦+𝑧>𝑥 x z y

4. Practice Can these measures be the sides of a triangle? 7, 5, 4
2, 1, 5
9, 6, 3
7, 8, 4

5. 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

6. 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?

7. 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 >𝑐 14>𝑐 𝑜𝑟 𝑐<14

8. 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. 6+𝑐>8 −6 −6 𝑐>2

9. 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: 2<𝑐<14 Therefore, any value between 2 and 14 (but not equal to 2 or 14) can be the length of the third side.

10. 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!)

11. 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 𝟒<𝒙<𝟖
9 and 11 𝟐<𝒙<𝟐𝟎
10 and 18 𝟖<𝒙<𝟐𝟖
What patterns do you see?

12. 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.”

13. 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.
The Triangle is…
Ambiguous
Unique
Non-Existent
Definition: Able to draw more than 1 triangle
Definition: Only able to draw 1 triangle
Definition: Not possible to draw a triangle
Use when given 3 angles that equal 180˚.
Use when given 3 side lengths the satisfy the Triangle Inequality Theorem.
Use when the two theorems WILL NOT work.

14. 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.

15. 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.

16. 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.

17. 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.

18. Practice
Given the following angle measurements, determine the third angle measurement.
60°, 80°
110°, 20°
Do these measurements create triangles?
3. 55°, 75°, 𝑎𝑛𝑑 50°
4. 80°, 90°, 𝑎𝑛𝑑 80°

19. Practice
Given the following angle measurements, determine the third angle measurement.
60°, 80° 𝟒𝟎°
110°, 20° 𝟓𝟎°
Do these measurements create triangles?
3. 55°, 75°, 𝑎𝑛𝑑 50° yes
4. 80°, 90°, 𝑎𝑛𝑑 80° no

20. 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?

21. 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?

22. 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.

23. Constructing Triangles from Angles
Conditions, such as angle measurements, that can create more than one triangle are called ambiguously defined.

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