1. 5.3 Trade Routes and Pasta Anyone?
Geometry
5.3 Trade Routes and Pasta Anyone?

2. 5.3: The Triangle Inequality Theorem
Objectives
Explore the relationships between the side lengths of a triangle and the measures of its interior angles.
Prove the Triangle Inequality Theorem

3. 3, 4, 5: Right Triangle: 5 2 =
6, 7, 8: Acute Triangle: 8 2 <
6, 8, 11: Obtuse Triangle: >
4, 6, 7: Acute Triangle: 7 2 <

4. Relationship Between Type of Triangle and Lengths of Sides
Acute Triangle
𝑐 2 < 𝑎 2 + 𝑏 2
Right Triangle
𝑐 2 = 𝑎 2 + 𝑏 2
Obtuse Triangle
𝑐 2 > 𝑎 2 + 𝑏 2

5. Problem 1: Who is Correct?
Together #1
Straw Activity Worksheet (5 Minutes)
Tear the straws to form the triangles
Answer the questions on the back
Investigation 1: What Is the Shortest Path from A to B?
The sum of the lengths of any two sides of a triangle is greater than the lengths of the third side.

6. Problem 1: Who is Correct?
#5-6 Together (Discuss)
#7 On Your Own (1 Minute)
2+2.4>5.1 Not a triangle
7+1.9>9.2 Not a triangle

7. Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the lengths of the third side. A B C
𝐴𝐵+𝐴𝐶>𝐵𝐶
𝐴𝐶+𝐵𝐶>𝐴𝐵
𝐵𝐶+𝐴𝐵>𝐴𝐶

8. Proof of Triangle Inequality Thm

9. Square root both sides (7)
Addition Property (9 & 10)
Segment Addition
Substitution (11 & 12)

10. Formative Assessment Skills Practice 5.3
Problem Set Pg (1-22)
Due Today
Reminder
The shortest side is opposite the smallest angle
The largest angle is opposite the longest side