lesson 5-2 inequalities and triangles. 5-minute check on lesson 5-1 transparency 5-2 in the figure,...

Lesson 5-2

Inequalities and Triangles

5-Minute Check on Lesson 5-15-Minute Check on Lesson 5-15-Minute Check on Lesson 5-15-Minute Check on Lesson 5-1 Transparency 5-2

In the figure, A is the circumcenter of LMN.

1. Find y if LO = 8y + 9 and ON = 12y – 11.

2. Find x if mAPM = 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 mRUS = 7y + 27.

5. Find RV if RV = 6a + 3 and RT = 10a + 14.

5-Minute Check on Lesson 5-15-Minute Check on Lesson 5-15-Minute Check on Lesson 5-15-Minute Check on Lesson 5-1 Transparency 5-2

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 mAPM = 7x + 13. 11

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 mRUS = 7y + 27. 9

5. Find RV if RV = 6a + 3 and RT = 10a + 14. 27

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

Vocabulary

• No new vocabulary words or symbols

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.

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

A

W

T

19

714

19 > 14 > 7

WT > AW > AT

A > T > W

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 m3 to m1.

By the Exterior Angle Theorem, m1 = m3 + m4. Since angle measures are positive numbers and

from the definition of inequality, m1 > m3.

Compare m4 to m1.

By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4.

Compare m5 to m1.

Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By substitution, m1 > m5.

By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2. Since we know that m1 > m5, by the Transitive Property, m1 > m2.

Compare m2 to m5.

Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure.

Answer: 1 has the greatest measure.

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.

EXAMPLE 2

Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14.

By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + m3.

Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7.

Answer: Thus, the measures of 4, 11, 9, 3, 2, 6, and 7 are all less than m14 .

EXAMPLE 3

Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5.

By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5.

Answer: Thus, the measures of 10, 16, 12, 15 and

17 are all greater than m5.

EXAMPLE 4

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 m4

b. all angles whose measures are greater than m8

Answer: 5, 2, 8, 7

Answer: 4, 9, 5

EXAMPLE 5

Determine the relationship between the measures of RSU and SUR.

Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

EXAMPLE 6

Determine the relationship between the measures of TSV and STV.

Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

EXAMPLE 7

Determine the relationship between the measures of RSV and RUV.

Answer: mRSV > mRUV

mRSU > mSUR

mUSV > mSUV

mRSU + mUSV > mSUR + mSUV

mRSV > mRUV

EXAMPLE 8

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

EXAMPLE 9

Summary & Homework

• Summary:– 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)