5'6 Indirect Proof and Inequalities in 2 s

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5.6 Indirect Proof and Inequalities in 2 ?s

• Pg 302

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Indirect proof

• Indirect proof a proof in which you prove that
  a statement is true by first assuming that the
  opposite is true.
• Steps
• Assume the prove statement is false
• Obtain statements that logically follow from your
  assumption
• If you find a contradiction, then the original
  prove stmt must have been true.

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Example Given ?ABCProve ?ABC doesnt have
more than one obtuse angle
B
A
C
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Indirect Proof

• 1. Start by assuming that ? ABC does have more
  than one obtuse ?
• 2. m?A gt90o and m?Bgt90o (def. of obtuse ?)
• m?A m?B gt 180o ( prop)
• However, you know that all 3 ?s must add up to
  180o (? sum thm)
• m? A m?B m?C 180o
• m?A m?B 180o m?C (- prop. )
• So, you can substitute 180o - m?C for m?A m?B in
  m?B m?A gt180
• 180o - m?Cgt180
• 0o gt m?C (- prop. )
• 3. The last statement isnt possible so you know
  that ? ABC cant have more than one obtuse ?.

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Thm 5.14Hinge thm

• If 2 sides of one ? are ? to the 2 sides of
  another ? and the included ? of the 1st ? is
  larger than the included ? of the 2nd ?, then the
  3rd side of the 1st ? is longer than the 3rd side
  of the 2nd ?.

If segAB ? segDE, seg BC ? seg EF, m?B gtm?E
then seg ACgt DF
D
A
60
85
E
F
B
C
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Thm 5.15Converse of the hinge thm

• If 2 sides of 1 ? are _at_ to 2 sides of another ? ,
  and the 3rd side of the 1st ? is larger than the
  3rd side of the 2nd ?, then the included ? of the
  1st is greater than the included ? of the 2nd.

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Example is (m?1) lt or gt (m?2)?

• gt

13
1
2
15
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Example Is AB gt or lt CB?
A
D

• gt

20o
B
18o
C
E
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Assignment