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Triangle Inequalities

Triangle Inequalities. § 7.1 Segments, Angles, and Inequalities. § 7.2 Exterior Angle Theorem. § 7.3 Inequalities Within a Triangle. § 7.4 Triangle Inequality Theorem. Vocabulary. Segments, Angles, and Inequalities. Inequalities. What You'll Learn.

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Triangle Inequalities

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

  2. Vocabulary Segments, Angles, and Inequalities Inequalities What You'll Learn You will learn to apply inequalities to segment and angle measures. 1) Inequality

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

  4. 60° 133° K J Segments, Angles, and Inequalities The measure of J is greater than the measure of K. inequalities The statements TU > VW and J > K are called __________ becausethey contain the symbol < or >. a < b a > b a = b

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

  6. C A B Segments, Angles, and Inequalities AB > CB AB > AC A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.

  7. D P E F Segments, Angles, and Inequalities A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.

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

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

  10. Segments, Angles, and Inequalities For any numbers a, b, and c, 1) if a < b, then a + c < b + c and a – c < b – c. 1 < 3 1 + 5 < 3 + 5 6 < 8 2) if a > b, then a + c > b + c and a – c > b – c. For any numbers a, b, and c,

  11. End of Section 7.1

  12. Vocabulary Exterior Angle Theorem What You'll Learn 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. P 1 3 4 2 Q R Exterior Angle Theorem interior In the triangle below, recall that 1, 2, and 3 are _______ angles ofΔPQR. exterior Angle 4 is called an _______ angle of ΔPQR. linear pair An exterior angle of a triangle is an angle that forms a _________ with one of the angles of the triangle. In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3. Remote interior angles ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

  14. 1 2 3 4 5 Exterior Angle Theorem In the figure below, 2 and 3 are remote interior angles with respect to what angle? 5

  15. X 1 2 3 4 Y Z Exterior Angle Theorem remote interior angles m4 = m1 + m2

  16. Exterior Angle Theorem

  17. X 1 2 3 4 Y Z Exterior Angle Theorem remote interior angles m4 > m1 m4 > m2

  18. 74° 2 1 3 Exterior Angle Theorem Name two angles in the triangle below that have measures less than 74°. 1 and3 acute

  19. Exterior Angle Theorem

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

  21. End of Section 7.2

  22. Vocabulary Inequalities Within a Triangle What You'll Learn You will learn to identify the relationships between the _____ and _____ of a triangle. sides angles Nothing New!

  23. P 11 8 M 13 L Inequalities Within a Triangle in the same order PM < ML LP < mM < mL < mP

  24. K W 45° 75° 60° J Inequalities Within a Triangle in the same order mW < mJ < mK KW < WJ JK <

  25. W X Y Inequalities Within a Triangle greatest measure 5 3 4 WY > XW WY > XY

  26. Inequalities Within a Triangle The longest side is The largest angle is So, the largest angle is So, the longest side is

  27. End of Section 7.3

  28. Vocabulary Triangle Inequality Theorem What You'll Learn You will learn to identify and use the Triangle Inequality Theorem. Nothing New!

  29. b a c Triangle Inequality Theorem greater a + b > c a + c > b b + c > a

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

  31. End of Section 7.4

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