Special Parallelograms

1 7.2 Special Parallelograms In the last section we defined parallelograms and discovered several properties that all parallelograms exhibit. In this section we will look at some special parallelograms which have some additional properties. These special parallelograms are rectangles, rhombuses, and squares. The definitions of these special parallelograms are listed below. A rectangle is a parallelogram with four right angles. Rectangle A rhombus is a parallelogram with four congruent sides. Rhombus 2 A square is a parallelogram with four congruent sides and four right angles. Square Notice that, by definition, a square is also a rectangle and a rhombus. So any special properties that a rectangle and a rhombus possess a square will possess as well. And since rectangles, rhombuses, and squares are also parallelograms they have all the properties of parallelograms that we learned about in the previous section. At the end of this section we will summarize all the properties that each of these special parallelograms exhibit. Let's look at some of them now. Theorem: The diagonals of a rectangle are congruent. Given: rectangle ABCD êêêêê êêêêê Prove: DB @ AC D C A B 3 Plan for proof: Since a rectangle is also a parallelogram opposite sides are congruent, so êêêêê êêêêê êêêêê êêêêê AD @ BC. —DAC and —CBA are congruent since they are both right angles. And AB @ AB so êêêêê êêêêê DDAB @ DCBA by SAS. So DB @ AC by CPCTC. The rest of the theorems are given without proof. Most of the proofs make use of congruent triangles and corresponding parts of congruent triangles. Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects two angles of the rhombus. The converses of the theorems above are also true and can be used to prove that a parallelogram is a rhombus or a rectangle. Theorem: If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 4 There is one other theorem of note that we need to look at which at first glance seems to have nothing to do with special parallelograms. However, the properties of rectangles lead to the following conclusion about any right triangle. Theorem: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Look at the figure and explanation below. X C M A B Begin with right DCAB. êêêêê 1. Draw lines to form rectangle CABX (draw an auxiliary line through C parallel to AB and êêêêê another through B parallel to AC. êêêêê 2. Draw AX. AX = CB since they are the diagonals of a rectangle. êêêêê êêêêê 3. AX and CB bisect each other since they are the diagonals of a parallelogram. 4. MC = MA = MB = MX. Since we have shown that MC = MA = MB we have that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. The table below summarizes the properties of parallelograms in general and the special properties of rectangles, rhombuses, and squares. 5 Property Parallelogram Rectangle Rhombus Square X X X X Opp. sides are @ X X X X Opp. —'s are @ X X X X Diags. bisect each other X X X X X X Opp. sides are »» All sides are @ Diags. are @ X X Diags. are ¦ X X A diag. bisects two —'s X X All —'s are rt. —'s X X