1. Splash Screen

2. Five-Minute Check (over Lesson 5–2) Then/Now
Key Concept: Definition of Inequality
Key Concept: Properties of Inequality for Real Numbers
Theorem 5.8: Exterior Angle Inequality
Example 1: Use the Exterior Angle Inequality Theorem
Theorems: Angle-Side Relationships in Triangles
Example 2: Identify Arithmetic Sequences
Example 3: Order Triangle Side Lengths
Example 4: Real-World Example: Angle-Side Relationships
Lesson Menu

3. Find the coordinates of the centroid of the triangle with vertices D(–2, 9), E(3, 6), and F(–7, 0).
B. (–3, 4)
C. (–2, 5)
D. (–1, 4) A B C D
5-Minute Check 1

4. In ΔRST, RU is an altitude and SV is a median
In ΔRST, RU is an altitude and SV is a median. Find y if mRUS = 7y + 27.
___
A. 5
B. 7
C. 9
D. 11 A B C D
5-Minute Check 3

5. A B C D In ΔRST, RU is an altitude and SV is a median.
___
Find RV if RV = 6a + 3 and RT = 10a + 14.
A. 3
B. 4
C. 21
D. 27 A B C D
5-Minute Check 4

6. Which of the following points is the center of gravity of a triangle?
A. centroid
B. circumcenter
C. incenter
D. orthocenter A B C D
5-Minute Check 5

7. You found the relationship between the angle measures of a triangle
You found the relationship between the angle measures of a triangle. (Lesson 4–2)
Recognize and apply properties of inequalities to the measures of the angles of a triangle.
Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.
Then/Now

8. Concept

9. Use the Exterior Angle Inequality Theorem
Example 1

10. Use the Exterior Angle Inequality Theorem
By the Exterior Angle Inequality Theorem, m14 > m4 and m14 > m11. In addition, m14 > m2 and m14 > m4 + m3, so m14 > m4 and m14 > m3.
Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7.
Example 1

11. Use the Exterior Angle Inequality Theorem
Example 1

12. Use the Exterior Angle Inequality Theorem
By the Exterior Angle Inequality Theorem, m10 > m5 and m16 > m10, so m16 > m5. Since 10 and 12 are vertical angles, m12 > m5. m15 > m12, so m15 > m5. In addition, m17 > m5 + m6, so m17 > m5.
Example 1

13. A B C D A. B. C. D.
Example 1

14. A B C D A. B. C. D.
Example 1

15. Concept

16. List the angles of ΔABC in order from smallest to largest.
Identify Arithmetic Sequence
List the angles of ΔABC in order from smallest to largest.
The sides from the shortest to longest are AB, BC, and AC. The angles opposite these sides are C, A, and B respectively. So, according to the Angle-Side Relationship, the angles from smallest to largest are C, A, B.
Answer: C, A, B
Example 2

17. A B C D List the angles of ΔTVX in order from smallest to largest.
A. X, T, V
B. X, V, T
C. V, T, X
D. T, V, X A B C D
Example 2

18. List the sides of ΔABC in order from shortest to longest.
Order Triangle Side Lengths
List the sides of ΔABC in order from shortest to longest.
The angles from smallest to largest are B, C, and A. The sides opposite these angles are AC, AB, and BC, respectively. So, the sides from shortest to longest are AC, AB, BC.
Answer: AC, AB, BC
Example 3

19. A B C D List the sides of ΔRST in order from shortest to longest.
A. RS, RT, ST
B. RT, RS, ST
C. ST, RS, RT
D. RS, ST, RT A B C D
Example 3

20. Angle-Side Relationships
HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?
Example 4

21. Answer: So, Ebony should tie the ends marked Y and Z.
Angle-Side Relationships
Theorem 5.10 states that if one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Since X is opposite the longest side it has the greatest measure.
Answer: So, Ebony should tie the ends marked Y and Z.
Example 4

22. KITE ASSEMBLY Tanya is following directions for making a kite
KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing? A B C D
A. A and D
B. B and F
C. C and E
D. A and B
Example 4

23. Homework p even, 22