2. Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 5-1Bisectors, Medians, and Altitudes Lesson 5-2Inequalities and Triangles Lesson 5-3Indirect Proof Lesson 5-4The Triangle Inequality Lesson 5-5Inequalities Involving Two Triangles

5. Lesson 2 Contents Example 1Compare Angle Measures Example 2Exterior Angles Example 3Side-Angle Relationships Example 4Angle-Side Relationships

6. Example 2-1a Determine which angle has the greatest measure. ExploreCompare the measure of  1 to the measures of  2,  3,  4, and  5. PlanUse properties and theorems of real numbers to compare the angle measures.

7. Example 2-1a 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.

8. 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. Example 2-1a Compare m  2 to m  5. ExamineThe 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.

9. Example 2-1b Determine which angle has the greatest measure. Answer:  5 has the greatest measure.

10. Example 2-2a 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.

11. Example 2-2b 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.

12. 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 Example 2-2c Answer:  5,  2,  8,  7 Answer:  4,  9,  5

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