1. Inequalities and Triangles
Lesson 5-2
Inequalities and Triangles

2. 5-Minute Check on Lesson 5-1
Transparency 5-2
5-Minute Check on Lesson 5-1
In the figure, A is the circumcenter of LMN.
1. Find y if LO = 8y + 9 and ON = 12y – 11.
2. Find x if mAPM = 7x + 13.
3. Find r if AN = 4r – 8 and AM = 3(2r – 11).
In RST, RU is an altitude and SV is a median.
4. Find y if mRUS = 7y + 27.
5. Find RV if RV = 6a + 3 and RT = 10a + 14.

3. 5-Minute Check on Lesson 5-1
Transparency 5-2
5-Minute Check on Lesson 5-1
In the figure, A is the circumcenter of LMN.
1. Find y if LO = 8y + 9 and ON = 12y – 11. 5
2. Find x if mAPM = 7x
3. Find r if AN = 4r – 8 and AM = 3(2r – 11). 12.5
In RST, RU is an altitude and SV is a median.
4. Find y if mRUS = 7y
5. Find RV if RV = 6a + 3 and RT = 10a

4. Objectives
Recognize and apply properties of inequalities to the measures of angles of a triangle
Recognize and apply properties of inequalities to the relationships between angles and sides of a triangle

5. Vocabulary
No new vocabulary words or symbols

6. Theorems
Theorem 5.8, Exterior Angle Inequality Theorem – If an angle is an exterior angle of a triangle, then its measure is greater that the measure of either of it corresponding remote interior angles.
Theorem 5.9 – 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.
Theorem 5.10 – If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

7. Key Concept
Step 1: Arrange sides or angles from smallest to largest or largest to smallest based on given information
Step 2: Write out identifiers (letters) for the sides or angles in the same order as step 1
Step 3: Write out missing letter(s) to complete the relationship
Step 4: Answer the question asked
19 > > 7
WT > AW > AT
A > T > W A W T 19 7 14

8. Determine which angle has the greatest measure.
Explore Compare the measure of 1 to the measures of 2, 3, 4, and 5.
Plan Use properties and theorems of real numbers to compare the angle measures.
Solve Compare m3 to m1.
By the Exterior Angle Theorem, m1 = m3 + m4. Since angle measures are positive numbers and from the definition of inequality, m1 > m3.
Compare m4 to m1.
By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4.

9. Compare m5 to m1.
Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By substitution, m1 > m5.
Compare m2 to m5.
By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2. Since we know that m1 > m5, by the Transitive Property, m1 > m2.
Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure.
Answer: 1 has the greatest measure.

10. EXAMPLE 2 Order the angles from greatest to least measure.
Answer: 5 has the greatest measure; 1 and 2 have the same measure; 4, and 3 has the least measure.

11. EXAMPLE 3
Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14.
By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + 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.
Answer: Thus, the measures of 4, 11, 9,  3,  2, 6, and 7 are all less than m14 .

12. EXAMPLE 4
Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5.
By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5.
Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

13. EXAMPLE 5
Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.
a. all angles whose measures are less than m4
b. all angles whose measures are greater than m8
Answer: 5, 2, 8, 7
Answer: 4, 9, 5

14. EXAMPLE 6
Determine the relationship between the measures of RSU and SUR.
Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

15. EXAMPLE 7
Determine the relationship between the measures of TSV and STV.
Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

16. EXAMPLE 8
Determine the relationship between the measures of RSV and RUV.
mRSU > mSUR
mUSV > mSUV
mRSU + mUSV > mSUR + mSUV
mRSV > mRUV
Answer: mRSV > mRUV

17. EXAMPLE 9
Determine the relationship between the measures of the given angles.
a. ABD, DAB
b. AED, EAD
c. EAB, EDB
Answer: ABD > DAB
Answer: AED > EAD
Answer: EAB < EDB

18. Summary & Homework Summary: Homework:
The largest angle in a triangle is opposite the longest side, and the smallest angle is opposite the shortest side
The longest side in a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle
Homework:
pg 251: (17-34, 46-50)