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The Converse of the Pythagorean Theorem

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... DEF with sides of length a and b, and hypotenuse of length x ... right triangles with one leg and hypotenuse congruent are congruent ... Hypotenuse-Leg ... – PowerPoint PPT presentation

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Title: The Converse of the Pythagorean Theorem


1
The Converse of the Pythagorean Theorem
  • A Pythagorean triple is a set of three integers
    which satisfy the Pythagorean Theorem
  • Some examples are
  • 3-4-5 5-12-13 7-24-25 8-15-17
  • Multiples of Pythagorean triples are also
    triples
  • 6-8-10 10-24-26 14-48-50 16-30-34
  • 9-12-15
  • 12-16-20
  • C-83 Converse of the Pythagorean Theorem
  • If the lengths of the three sides of a triangle
    satisfy the Pythagorean equation, then the
    triangle is a right triangle

hypotenuse
legs
2
The Converse of the Pythagorean Theorem
  • Proving the Converse of the Pythagorean Theorem
  • Given ?ABC with side lengths such that a2 b2
    c2
  • Construct right ?DEF with sides of length a and
    b, and hypotenuse of length x
  • By the Pythagorean Theorem, a2 b2 x2
  • By substitution, x2 c2, so x c
  • The two triangles are congruent by SSS
  • Angle C must be a right angle by CPCTC
  • Therefore ?ABC is a right triangle, QED

3
Right Triangle Congruence
  • Prove that two right triangles with one leg and
    hypotenuse congruent are congruent
  • Given right ?ABC and right ?DEF with legs of
    length a and hypotenuse of length c
  • Is there a congruence rule that proves ?ABC _at_
    ?DEF?
  • The matching parts would be SSA, which is not a
    congruence rule
  • By the Pythagorean Theorem, a2 b2 c2, so the
    other legs of the two triangles must also be
    congruent
  • Therefore the triangles are congruent by SSS
  • Hypotenuse-Leg Congruence Theorem
  • If one leg and the hypotenuse of one right
    triangle is congruent to one leg and the
    hypotenuse of a second right triangle, then the
    triangles are congruent

c
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