Inequalities and Triangles Lesson 5-2 Inequalities and Triangles

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.

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 + 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 mRUS = 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 19 > 14 > 7 WT > AW > AT A > T > W A W T 19 7 14

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.

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.

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.

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 .

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.

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

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.

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.

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

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

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)