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Triangle Inequalities

inequaltites in a triangle and theorems

kimginer
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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. 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.

  3. 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,

  4. 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

  5. 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.

  6. 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

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

  8. Exterior Angle Theorem

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

  10. 74° 2 1 3 Exterior Angle Theorem Name two angles in the triangle below that have measures less than 74°. 1 and3 acute

  11. Exterior Angle Theorem

  12. ? 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

  13. 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!

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

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

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

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

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

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

  20. 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

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