1.  § 7.1 Segments, Angles, and Inequalities  § 7.4 Triangle Inequality Theorem  § 7.3 Inequalities Within a Triangle  § 7.2 Exterior Angle Theorem

2. You will learn to apply inequalities to segment and angle measures. 1) Inequality Inequalities

3. The Comparison Property of Numbers is used to compare two line segments of unequal measures. The property states that given two unequal numbers a and b, either: a < b or a > b The same property is also used to compare angles of unequal measures. T U 2 cm V W 4 cm The length of is less than the length of, or TU < VW

4. J 133° K 60° The measure of  J is greater than the measure of  K. The statements TU > VW and  J >  K are called __________ because they contain the symbol. inequalities Postulate 7 – 1 Comparison Property For any two real numbers, a and b, exactly one of the following statements is true. a < b a = b a > b

5. 6 4 2 0 -2 S D N Replace with, or = to make a true statement. SN DN 6 – (- 1) 6 – 2 7 4 > > Lesson 2-1 Finding Distance on a number line.

6. Theorem 7 – 1 If point C is between points A and B, and A, C, and B are collinear, then ________ and ________. A C B AB > AC AB > CB A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate.

7. Theorem 7 – 2 D P F E A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate.

8. 108° 149° 45° 40° 18° A B C D Replace with, or = to make a true statement. m  BDA m  CDA 45°40° + 45 ° < < Use theorem 7 – 2 to solve the following problem. Check: 45°85 °

9. Property Transitive Property For any numbers a, b, and c, 1) if a < b and b < c, then a < c. 2) if a > b and b > c, then a > c. if 5 < 8 and 8 < 9, then 5 < 9. if 7 > 6 and 6 > 3, then 7 > 3.

10. Property Addition and Subtraction Properties Multiplication and Division Properties For any numbers a, b, and c, 1) if a < b, then a + c < b + c and a – c < b – c. 2) if a > b, then a + c > b + c and a – c > b – c. 1 < 3 1 + 5 < 3 + 5 6 < 8

12. You will learn to identify exterior angles and remote interior angles of a triangle and use the Exterior Angle Theorem. 1) Interior angle 2) Exterior angle 3) Remote interior angle

13. 1 2 34 P QR In the triangle below, recall that  1,  2, and  3 are _______ angles of ΔPQR. interior Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________ with one of the angles of the triangle. linear pair In ΔPQR,  4 is an exterior angle at R because it forms a linear pair with  3. ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR,  1, and  2 are the remote interior angles with respect to  4.

14. 1 2 345 In the figure below,  2 and  3 are remote interior angles with respect to what angle? 55

15. Theorem 7 – 3 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to sum of the measures of its ___________________. remote interior angles X 432 1 ZY m  4 =m  1 + m  2

17. Theorem 7 – 4 Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measures of either of its two ____________________. remote interior angles X 432 1 ZY m  4 >m1m1 m2m2

18.  1 and  3 74° 13 2 Name two angles in the triangle below that have measures less than 74°. Theorem 7 – 5 If a triangle has one right angle, then the other two angles must be _____. acute

20. The feather–shaped leaf is called a pinnatifid. In the figure, does x = y? Explain. x = y ? __ + 81 = 32 + 78 28 28° 109 = 110 No! x does not equal y

22. You will learn to identify the relationships between the _____ and _____ of a triangle. sides angles Nothing New!

23. Theorem 7 – 6 If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal ________________. 13 8 11 L P M in the same order LP < PM <ML m  M < mPmPm  L <

24. Theorem 7 – 7 If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal ________________. in the same order JK < KW <WJ m  W < mKmKm  J < J 45° W K 60° 75°

25. Theorem 7 – 8 In a right triangle, the hypotenuse is the side with the ________________. greatest measure WY > XW 3 5 4 Y W X WY > XY

26. The longest side is So, the largest angle is The largest angle is So, the longest side is

28. You will learn to identify and use the Triangle Inequality Theorem. Nothing New!

29. Theorem 7 – 9 Triangle Inequality Theorem The sum of the measures of any two sides of a triangle is _______ than the measure of the third side. greater a b c a + b > c a + c > b b + c > a

30. Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10 However, 10 + 5 > 16