1. ESSENTIAL QUESTION: How to use triangle measurements to decide which side is longest and which angle is largest?

2. Theorem 4.10
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

3. Name the angles from largest to smallest.
Example 1
Order Angle Measures
Name the angles from largest to smallest.
SOLUTION
TV > TU, so mU > mV. Also, TU > UV, so mV > mT.
ANSWER
The order of the angles from largest to
smallest is U, V, T. 3

4. Name the angles from largest to smallest.
Checkpoint
Order Angle Measures and Side Lengths
Name the angles from largest to smallest. 1. 2. 3.

5. Name the angles from largest to smallest.
Checkpoint
Order Angle Measures and Side Lengths
Name the angles from largest to smallest. 1.
ANSWER
N; L; M 2.
ANSWER
Q; R; P 3.
ANSWER
U; S; T

6. Theorem 4.11
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

7. Name the sides from longest to shortest.
Example 2
Order Side Lengths
Name the sides from longest to shortest.
SOLUTION
mE > mD, so DF > FE. Also, mD > mF, so FE > DE.
ANSWER
The order of the sides from longest to
shortest is DF, FE, DE. 7

8. Name the sides from longest to shortest.
Checkpoint
Order Angle Measures and Side Lengths
Name the sides from longest to shortest. 4. 5. 6.

9. Name the sides from longest to shortest.
Checkpoint
Order Angle Measures and Side Lengths
Name the sides from longest to shortest. 4.
ANSWER
GH; JG; JH 5.
ANSWER
DE; EF; DF 6.
ANSWER
AC; AB; BC

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

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

12. Can the side lengths form a triangle? Explain.
Example 3
Use the Triangle Inequality
Can the side lengths form a triangle? Explain.
3, 5, 9 a.
3, 5, 8 b.
3, 5, 7 c.
SOLUTION a. b. c.
These lengths do not form a triangle, because < 9.
These lengths do not form a triangle, because = 8.
These lengths do form a triangle, because > 7, > 5, and > 3. 12

13. Can the side lengths form a triangle? Explain.
Checkpoint
Use the Triangle Inequality
Can the side lengths form a triangle? Explain. 7.
5, 7, 13 8.
6, 9, 12 9.
10, 15, 25

14. Can the side lengths form a triangle? Explain.
Checkpoint
Use the Triangle Inequality
Can the side lengths form a triangle? Explain. 7.
5, 7, 13
ANSWER
No; < 13.
ANSWER
Yes; > 12, > 9, and > 6. 8.
6, 9, 12 9.
10, 15, 25
ANSWER
No; = 25.

15. 1.
BD is a median of ∆ABC. Find the length of AD.
ANSWER 7 2.
Point P is the centroid of ∆LMN
and QN = 45. Find PN and QP.
ANSWER
PN = 30, QP = 15

16. 3.
Point D is the centroid of ∆ABC and DE = 14. Find CD and CE.
ANSWER
CD = 28, CE = 42

17. Hw Practice 4.7A