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Lesson 5-3

Lesson 5-3. Inequalities and Triangles. Transparency 5-2. 5-Minute Check on Lesson 5-1. In the figure, A is the circumcenter of  LMN . 1. Find y if LO = 8 y + 9 and ON = 12 y – 11. 2. Find x if m  APM = 7 x + 13. 3. Find r if AN = 4 r – 8 and AM = 3(2 r – 11).

shantell-allen
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Lesson 5-3

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  1. Lesson 5-3 Inequalities and Triangles

  2. 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. 6. Which congruence statement is true if P is the circumcenter of WXY? Standardized Test Practice: WPWX WYXY C D WPXP A WXXY B

  3. 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 6. Which congruence statement is true if P is the circumcenter of WXY? Standardized Test Practice: WPWX WYXY C D WPXP A WXXY B

  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 > 14 > 7 WT > AW > AT A > T > W A 7 14 T 19 W

  8. By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4. 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.

  9. 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. 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5 to m1. Compare m2 to m5. 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. 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. 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. 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, andm12 > m5, som15 > m5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

  13. 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. Determine the relationship between the measures of RSUand SUR. Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

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

  16. Determine the relationship between the measures of RSVand 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. 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: • 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 346-47: 8-13, 15, 17, 31, 33, 35

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