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How do you prove "if and only if" statements?

Evan really enjoyed his geometry assignment! Furthermore, he was learning so much as he researched parallelograms in order to write a compelling story. Evan continued...

Detective Reese proved that in a parallelogram, opposite angles are congruent and adjacent angles are supplementary. Now, he needed to prove that diagonals in a rectangle are congruent. Therefore, Detective Reese examined several different rectangle parallelograms. After a thorough investigation, he noticed something peculiar: Every parallelogram that is a rectangle has congruent diagonals. AND every parallelogram with congruent diagonals is a rectangle. This statement requires a special type of proof, called if and only if. Specifically, a parallelogram is a rectangle if and only if its diagonals are congruent. (Sometimes, you'll see if and only if abbreviated as iff.)

Before Detective Reese could prove this particular parallelogram postulate, he had to consider exactly how to prove iff statements. These statements are special because each one actually requires two proofs. To help determine what proofs were needed, Reese used the examples in the tab set below.

Example 1

Example 2

Example 3

Example 4

Geometry instrumentsAssume you want to prove that a student is in geometry if and only if she has already taken algebra. Then, you have to prove these two statements:

  1. If a student is in geometry, then she has taken algebra.
  2. If a student has taken algebra, then she is in geometry.

Keep in mind, this statement may or may not be true. It's just an example of how you go about proving iff statements.

Rose

Assume you want to prove that a plant is a rose if and only if it's a flower. Then, you have to prove these two statements:

  1. If a plant is a rose, then it's a flower.
  2. If a plant is a flower, then it's a rose.

Again, keep in mind, this statement may or may not be true. It's just another example of how you go about proving if statements.

circles

Next, consider a geometric example. Assume you want to prove that a figure is a circle if and only if the distance around the figure is twice the product of pi and the radius. Then, you need to prove these two statements:

  1. If a figure is a circle, then the distance around it is twice the product of pi and the radius.
  2. If the distance around a figure is twice the product of pi and the radius, then the figure is a circle.

Now, consider the parallelogram postulate: A parallelogram is a rectangle if and only if its diagonals are congruent. What two statements do you need to prove in order to show that this postulate is true?

  1. If a parallelogram is a rectangle, then its diagonals are congruent.
  2. If a parallelogram has congruent diagonals, then it is a rectangle.

Question

How do you prove this statement: A parallelogram is a rectangle iff its diagonals are congruent?

Prove these two statements: (1) If a parallelogram is a rectangle, then its diagonals are congruent. (2) If the diagonals are congruent, then the parallelogram is a rectangle.