Lesson 5-2

1. Lesson 5-2 Inequalities and Triangles

2. Ohio Content Standards:

3. Ohio Content Standards: Formally define geometric figures.

4. Ohio Content Standards: Formally define and explain key aspects of geometric figures, including:a. interior and exterior angles of polygons;b. segments related to triangles (median, altitude, midsegment);c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);

5. Ohio Content Standards: Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

6. Ohio Content Standards: Solve equations and inequalities having rational expressions as coefficients and solutions.

7. Definition of Inequality

8. Definition of Inequality For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c.

9. Definition of Inequality For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c. Example: If 6 = 4 + 2, 6 > 4 and 6 > 2

10. Determine which angle has the greatest measure. 5 4 3 2 1

11. Theorem 5.8Exterior Angle Inequality Theorem

12. Theorem 5.8Exterior Angle Inequality Theorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles.

13. Theorem 5.8Exterior Angle Inequality Theorem B 2 1 3 4 A C

14. Use the Exterior Angle Inequality to list all of the angles that satisfy the stated condition. 14 17 15 16 5 4 6 3 11 10 12 9 2 7 1 8

15. Use the Exterior Angle Inequality to list all of the angles that satisfy the stated condition. all angles whose measures are less than m 14 14 17 15 16 5 4 6 3 11 10 12 9 2 7 1 8

16. Use the Exterior Angle Inequality to list all of the angles that satisfy the stated condition. all angles whose measures are greater than m 5 14 17 15 16 5 4 6 3 11 10 12 9 2 7 1 8

17. Theorem 5.9

18. 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.

19. Determine the relationship between the measures of the given angles. 5.2 R S 6.6 T 3.6 4.8 5.3 4.4 U 5.1 V

20. Determine the relationship between the measures of the given angles. 5.2 R S 6.6 T 3.6 4.8 5.3 4.4 U 5.1 V

21. Determine the relationship between the measures of the given angles. 5.2 R S 6.6 T 3.6 4.8 5.3 4.4 U 5.1 V

22. Determine the relationship between the measures of the given angles. 5.2 R S 6.6 T 3.6 4.8 5.3 4.4 U 5.1 V

23. Theorem 5.10

24. 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 that the side opposite the lesser angle.

25. Ebonyis following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie? 22.6 Y Z 16 16 X

26. Assignment:Pgs. 252-254 18-34 evens, 38-42 evens