# Thread: Parallelogram proof (is my work here correct)?

1. ## 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?

2. ## 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