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5-6 Inequalities in One Triangle

5-6 Inequalities in One Triangle. Properties of Inequalities. Previously we dealt with congruent segments and angles (having equal lengths or measure) We used the Properties of Equality and Congruence Now we will deal with segments of unequal lengths and angles with unequal measures

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5-6 Inequalities in One Triangle

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  1. 5-6 Inequalities in One Triangle

  2. Properties of Inequalities • Previously we dealt with congruent segments and angles (having equal lengths or measure) • We used the Properties of Equality and Congruence • Now we will deal with segments of unequal lengths and angles with unequal measures • We will use the Properties of Inequality and apply them to segment lengths and angle measures

  3. Properties of Inequalities • We will use these properties with the understanding that they apply only to positive numbers. • Think positive lengths and angle measurements • Most of the properties that we will use are based on one main principle: the angle/segment addition property • Two (or more) parts added together make the whole.

  4. Example 1 • Given: AC > AB and AB > BC • Conclusion: AC _____ BC • Assign values for each segment length if it helps • AC > BC B A C

  5. Properties of Inequalities- Transitive Property • If a>b and b>c, then a>c • Example: If AC>AB and AB>BC, then AC>BC • Example: If A > B, and B > C, then A > C

  6. Example 2- Using the Angle Addition Postulate • Given: • Conclusion: • Again, plug in values if it helps • The total is greater than each of the parts B C A D

  7. Properties of Inequalities- Comparison Property of Inequalities If a = b + c, and c > 0, then a > b. • Example: If x = y + 1, then x>y and x>1 (as long as x and y are greater than 0) • Example: If XY + YZ = XZ, then XY < XZ and YZ < XZ • Segment Addition Postulate • This tells us that if 2 parts add up to the whole, then the whole must be larger than the two parts.

  8. Properties of Equalities- reviewExterior Angle Theorem • The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles. • <BCD is the remote angle and <A and <B are the two remote interior angles • Since AB || CE, <B ≅<1 and <A ≅ <2 • m<BCD = m<1 + m<2 or m<BCD = m<A + m<B E B 1 2 A D C

  9. Corollary to the Triangle Exterior Angle Theorem • The measure of an exterior angle of a triangle will be greater than the measure of each of the two remote interior angles. • Since m<A + m<B = m<BCD, then • Again, this tells us that if 2 parts add up to the whole, then the whole must be larger than the two parts. B A D C

  10. Properties of Inequalities • If we have 2 segments: • MO with midpoint N, and QS with midpoint R • MO > QS • Tell if we know the following are true: • 1) MN > RS • 2) 2MN = MO • 3) 2QR > NO • 4) QR + RS = QS M N Q R S O

  11. Side Lengths & Angle Lengths • We have seen in isosceles triangles that we have two congruent sides. • What do we know about their opposite angles?

  12. Side Lengths & Angle Lengths • In any triangle, however, the same principle is at work. • If two congruent sides have congruent opposite angles, then two non-congruent sides will have two non-congruent opposite angles.

  13. Theorem 5-10 • If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

  14. Theorem 5-11 • If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

  15. Side Lengths & Angle Measures • In a triangle, the angle opposite the largest side is the largest angle and vice versa. • Also, the angle opposite the smallest side is the smallest angle and vice versa.

  16. Triangle Inequality Theorem • In a triangle, the three sides have specific relationships and ratios that exist in every triangle. • Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  17. Triangle Inequality Theorem • Are the following sets of values possible lengths for a triangle? • 1) 2, 5, 6 • 2) 6, 11, 12 • 3) 15, 15, .00001 • 4) 7, 9, 21

  18. Triangle Inequality Theorem • Given the first two values, what is the possible range of side for the third? • 5, 8, ??? = How can we find this set of values?

  19. Triangle Inequality Theorem • Given the first two values, what is the possible range of side for the third? • 1) 9, 12, ___ • 2) 1, 5, ___ • 3) 18, 22, ___

  20. Classwork • 5-6 worksheet

  21. Homework • 5-6 worksheet (what is not completed in class)

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