Unit 1, Lesson 13
Proofs about Parallelograms
Integrated Math 2
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Here is parallelogram and rectangle . What do you notice? What do you wonder?
Parallelogram A B C D, with A B parallel to C D and A D parallel to B C. Diagonal line segments A C and B D are drawn, intersecting at point X. Rectangle E F G H, with E F parallel to G H and F G parallel to E H. Diagonal line segments H F and E G are drawn, intersecting at point Y.
Prove: is a rectangle (angles and are right angles).
With your partner, you will work backward from the statement to the proof until you feel confident that you can prove that is a rectangle using only the given information.
Start with this sentence: I would know is a rectangle if I knew .
Then take turns saying this sentence: I would know [what my partner just said in the blank] if I knew .
Write down what each of you say. If you get to a statement and get stuck, go back to an earlier statement and try to take a different path.
A quadrilateral is a parallelogram if and only if its diagonals bisect each other. The “if and only if” language means that both the statement and its converse are true. So we need to prove:
To prove part 1, make the statement specific: If quadrilateral has diagonals and that intersect at such that is congruent to and is congruent to , then side is parallel to side , and side is parallel to side .
We could prove triangles and are congruent by the Side-Angle-Side Triangle Congruence Theorem. That means that corresponding angles in the triangles are congruent, so angle is congruent to . This means that alternate interior angles formed by lines and are congruent, so lines and are parallel. We could also make an argument that shows triangles and are congruent. Then, angles and are congruent, which means that lines and must be parallel.
To prove part 2, make the statement specific: If parallelogram has side parallel to side and side parallel to side , and diagonals and that intersect at , then we are trying to prove that is the midpoint of and of .
We could use a transformation proof. Rotate parallelogram by using the midpoint of diagonal as the center of the rotation. Then show that the midpoint of diagonal is also the midpoint of diagonal . That point must be since it is the only point on both line and line . So must be the midpoints of both diagonals, meaning the diagonals bisect each other.
We have proved that any quadrilateral with diagonals that bisect each other is a parallelogram, and that any parallelogram has diagonals that bisect each other. Therefore, a quadrilateral is a parallelogram if and only if its diagonals bisect each other.