Geometry: The Most Popular Parallelograms
The Most Popular Parallelograms
Rectangles, rhombuses, and squares are three specific kinds of parallelograms. They all have the properties of a parallelogram: Their opposite sides are parallel, their diagonals bisect each other and divide the parallelogram into two congruent triangles, and opposite sides and angles are congruent. But rectangles, rhombuses, and squares have additional characteristics that other parallelograms don't have.
A rectangle is a parallelogram that has a right angle. Actually, from this little bit of information, you know about all four angles of a rectangle. A rectangle is a parallelogram, so its opposite angles are congruent and its consecutive angles are supplementary. Recall that the supplement of a right angle is another right angle. So a rectangle actually has four right angles.
A rectangle is a parallelogram that has a right angle.
Rectangles have some properties that generic parallelograms do not. One such property is that the diagonals of a rectangle are congruent. I will state that as a theorem and discuss a game plan for the proof. I will leave the details up to you.
- Theorem 15.9: The diagonals of a rectangle are congruent.
To prove this theorem, take a look at the rectangle in Figure 15.9. Rectangle ABCD has diagonals ¯AC and ¯BD. In order to prove that they are congruent, you will want to use CPOCTAC. But which two triangles do you show are congruent? I would recommend that you show ΔADC ~= ΔDAB. Split these two triangles apart and make the connections. They are both right triangles, so that's one pair of congruent angles. Because opposite sides are congruent, you will be able to use our SAS Postulate to show ΔADC ~= ΔDAB.
A rhombus is a parallelogram with two congruent adjacent sides. Just as you saw with a rectangle, a rhombus inherits all of the desirable properties of a parallelogram. And rhombuses have special properties that generic parallelograms and rectangles do not have. I'll write these special properties as theorems that you can refer to later.
The first property of a rhombus is that all sides of a rhombus are congruent. That is not surprising, because you already know that the opposite sides of a rhombus are congruent (because it's a parallelogram). If opposite sides and adjacent sides are congruent, they are all congruent. Not all parallelograms and rectan gles have this special property.
A rhombus is a parallelogram with two congruent adjacent sides.
- Theorem 15.10: All sides of a rhombus are congruent.
The next property of a rhombus worth mentioning is that its diagonals are perpendicular. That one might not be so obvious, and is worth writing out a formal proof.
- Theorem 15.11: The diagonals of a rhombus are perpendicular.
- Example 4: Write a formal proof of Theorem 15.11.
- Solution: As usual, I'll write the formal proof step-by-step. Figure 15.10 shows the rhombus ABCD with diagonals ¯AC and ¯BD that intersect at M.
- Given: Rhombus ABCD with diagonals ¯AC and ¯BD.
- Prove: ¯AC ⊥ ¯BD.
- Proof: You need a game plan. Hang on to your hat because this one is pretty involved. In order to show that ¯AC ⊥ ¯BD, you need to show that ¯AC and ¯BD intersect and form a right angle. So you need to show that ∠AMB is a right angle. Because ∠AMB and ∠BMC form a straight angle, they are supplementary. If you could show that they were congruent, then they would have to be right angles. The easiest way to show ∠AMB ~= ∠BMC is to show ΔAMB ~= ΔCMB and use CPOCTAC. Now, because you're dealing with a rhombus, ¯AB ~= ¯BC. By the reflexive property of ~=, you know ¯BM ~= ¯BM. Because a rhombus is a parallelogram, and showed that the diagonals of a parallelogram bisect each other, ¯AM ~= ¯MC. You can use our SSS Postulate to conclude that ΔAMB ~= ΔCMB. Let's write it up.
|1.||Parallelogram ABCD has diagonals ¯AC and ¯BD that intersect at M||Given|
|2.||¯AM ~= ¯MC||Theorem 15.6|
|3.||¯AB ~= ¯BC||Definition of a rhombus|
|4.||¯BM ~= ¯BM||Reflexive property of ~=|
|5.||ΔAMB ~= ΔCMB||SSS Postulate|
|6.||∠AMB ~= ∠BMC||CPOCTAC|
|7.||∠AMB and ∠BMC form a straight angle||Definition of straight angle|
|8.||m∠AMB + m∠BMC = 180º||Angle Addition Postulate|
|9.||m∠AMB + m∠AMB = 180º||Substitution (steps 6 and 8)|
|10.||∠AMB is right||Algebra|
|11.||∠AMB is right||Definition of right angle|
|12.||¯AC ⊥ ¯BD||Definition of ⊥|
A square is both a rectangle and a rhombus. A square can be defined as a rectangle with congruent adjacent sides, or it could be defined as a rhombus that has a right angle. I'll pick the former description as the official definition. A square inherits all of the properties of a parallelogram, a rectangle, and a rhombus. A square has the best of all worlds. It has the properties of a parallelogram (opposite sides congruent, opposite angles congruent, opposite sides parallel, and diagonals bisect each other), a rectangle (diagonals are congruent and all four angles are congruent) and a rhombus (diagonals are perpendicular and all four sides are congruent).
A square is a rectangle with congruent adjacent sides.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.