B. Definition of an inequality

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B. Definition of an inequality

• For any real numbers a and b, a gt b if and only
  if there is a positive number c such that a b
  c.

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Properties of Inequalities for Real Numbers p.
254 in book

• Comparison Property
• one of three things has to be true
• altb agtb ab
• Transitive Property
• If altb and bltc, then altc
• If agtb and bgtc, then agtc

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• Addition and Subtraction Property
• if agtb then acgtbc and a-cgtb-c
• if altb then acltbc and a-cltb-c
• Multiplication and Division Property
• if cgto (positive)and agtb then acgtbc and
  a/cgtb/c
• if cgto (positive)and altb then acltbc and
  a/cltb/c
• if clto (negative) and agtb then acltbc and
  a/cltb/c
• if clto (negative) and altb then acgtbc and
  a/cgtb/c
• REMEMBERif you multiply or divide an inequality
  by a negative number it switches the direction of
  the inequality

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III. Examples

• 1.  Which assumption would you make to start an
  indirect proof of the statement two acute angles
  are congruent.

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• 2.   Which assumption would you make to start an
  indirect proof of the following statements?
• Bob took the dog for a walk
• EF is not a perpendicular bisector
• 3x 4 y 1
• lt 1 is less than or equal to lt 2

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• 3. Name the property that justifies if a lt b,
  then a c lt b c.
• 4. Name the property that justifies that if a is
  less than b, then a cannot be greater than b.
• 5. Given 2 y 8 16
• Prove y ? 5

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• 6. Given JKL with side lengths 3, 4, 5
• Prove lt K lt lt L

J
K
L
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Refer to page 255 in your textbook and try 5
13(hint for 8 10 use the exterior angle
inequality theorem-the exterior angle is greater
than the remote interior angles)
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Answers to p 255-2565 -13

• 5. Assume lines l and m do not intersect at x
• 6. Assume If the alt int lts formed by two
  parallel lines
• and a transversal are congruent, then the
  lines are
• not parallel.
• 7. Assume Sabrina did not eat the leftover pizza
• 8. lt3, lt7, lt5, lt6
  9. lt4 and lt8
• 10. lt8 gt lt7 because lt7 is part of a remote
  interior angle for the exterior lt8
• 11. Division 12. Addition 13.
  Transitive

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5-4 The Triangle Inequality

• I. Theorem
• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side.

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II. Examples

• 1. If Mrs. Ewing gave Elizabeth four pieces of
  tubing measuring 6 m, 7 m, 9 m, and 16 m, how
  many different triangles could she make?

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• 2. What are the possible lengths for the third
  side of a triangle with two sides of 8 and 13?

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• 3. How many triangles can be made from a rope 10
  ft long?

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5-5 Inequalities with two triangles

• I. Theorem
• SAS Inequality

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Theorem

• SSS Inequality

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5-2 Right Triangles

• I. What are the postulates for proving 2 right
  triangles are congruent?

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SAS aka LL
LL means Leg Leg
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ASA, AAS aka LA
LA means Leg Angle
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AAS aka HA
HA means Hypotenuse Angle
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Finally, a new one!

• HL (only for right triangles/in non-right
  triangles SSA)

HL means Hypotenuse Leg
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1. In the figure,      is the angle bisector lt BAC. Are the triangles congruent? 1. In the figure,      is the angle bisector lt BAC. Are the triangles congruent?
                                 
Tell which of the right triangle methods will
prove the triangles congruent.
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2. Find x so that the right triangles are
congruent.
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3. What do you need to prove by HA?