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Math Help - Parallelogram proof (is my work here correct)?

  1. #1
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    Parallelogram proof (is my work here correct)?

    One theorem states that if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. I did the following two column proof below to show this. Is it correct?

    Suppose we have parallelogram ABCD where A = lower left vertex, B = upper left vertex, C = upper right vertex, and D = lower right vertex. Suppose A is the right angle given.

    Statement/Reason
    1. ABCD is a parallelogram/Given
    2. m∠A = 90/Given
    3. m∠A ≅ m∠C/Definition of parallelogram
    4. m∠A + m∠B = 180/Same side interior angles are supplementary.
    5. m∠A ≅ m∠B/Subtraction property of equality
    6. m∠B ≅ m∠D/Definition of parallelogram
    7. ABCD is a rectangle/Definition of rectangle

    The one I'm really worrying about is #5. Can I use subtraction property of equality to justify that? Or do I just use the given?
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  2. #2
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    Re: Parallelogram proof (is my work here correct)?

    One theorem states that if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. I did the following two column proof below to show this. Is it correct?

    Suppose we have parallelogram ABCD where A = lower left vertex, B = upper left vertex, C = upper right vertex, and D = lower right vertex. Suppose A is the right angle given.

    Statement/Reason
    1. ABCD is a parallelogram/Given
    2. m∠A = 90/Given
    3. m∠A ≅ m∠C/Definition of parallelogram
    4. m∠A + m∠B = 180/Same side interior angles are supplementary.
    5. m∠A ≅ m∠B/Subtraction property of equality
    6. m∠B ≅ m∠D/Definition of parallelogram
    7. ABCD is a rectangle/Definition of rectangle

    The one I'm really worrying about is #5. Can I use subtraction property of equality to justify that? Or do I just use the given?

    A few comments and then my own solution:
    First, I would combine steps 1 and 2, the given information.
    Step 3 is not the definition of a parallelogram; it is a property of a parallelogram.

    This is what I would do:

    1. ABCD is a parallelogram, m∠A = 90 (Given)
    2. m∠A ≅ m∠C (Opposite angles of a parallelogram are congruent)
    3. m∠A + m∠B = 180 (Same side interior angles are supplementary)
    4. m∠B = 90 (Substitution and subtraction property of equality)
    5. m∠B = m∠D = (Opposite angles of a parallelogram are congruent)
    6. m∠A = m∠B = m∠C = m∠D = 90 (Steps 1, 2, 4 and 5)
    7. ABCD is a rectangle (Definition of rectangle,i.e., quadrilateral with 4 R angles)

    Good luck!
    -Andy
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