1. 1. Obj. 26 Special Parallelograms The student is able to (I can): Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems. Prove that a given quadrilateral is a rectangle, rhombus, or square.
2. 2. rectangleA parallelogram with four right angles.If a parallelogram is a rectangle, then its diagonals are congruent (checking for square). FIFS IH HS
3. 3. Because a rectangle is a parallelogram, it also inherits all of the properties of a parallelogram: Opposite sides parallel Opposite sides congruent Opposite angles congruent (actually all angles are congruent) Consecutive angles supplementary Diagonals bisect each other
4. 4. ExampleFind each length. 1. LW LW = FO = 30F30O 17L 2. OL OL = FW = 2(17) = 343. OW OWL is a right triangle, so OW 2 + LW 2 = OL2 OW 2 + 302 = 34 2 OW 2 + 900 = 1156 OW 2 = 256 OW = 16W
5. 5. rhombusA parallelogram with four congruent sides.If a parallelogram is a rhombus, then its diagonals are perpendicular.
6. 6. Proof:BO SLWBecause BOWL is a rhombus, BO OW. Diagonals bisect each other, so BS WS. The reflexive property means that OS OS. Therefore, OSB OSW by SSS. This means that OSB OSW. Since they are also supplementary, they must be 90.
7. 7. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. 31 281 2 3 4 5 6 7 87465Since opposite angles are also congruent: 1 2 5 6 3 4 7 8
8. 8. Examples1. What is the perimeter of a rhombus whose side length is 7? 4(7) = 28 2. Find the value of x The side = 10 Pyth. triple: 6, 8, 10 x=6Perimeter = 40(13y9)3. Find the value of y 13y 9 = 3y + 11 10y = 20 y=2x10 8(3y+11)
9. 9. squareA quadrilateral with four right angles and four congruent sides.Note: A square has all of the properties of both a rectangle and a rhombus: Diagonals are congruent Diagonals are perpendicular Diagonals bisect opposite angles.
10. 10. Conditions for Special ParallelogramsYou can always use the definitions to prove these, but there are also some shortcuts we can use. For all of these shortcuts, we must first prove or know that the quadrilateral is a parallelogram. To prove a parallelogram is a rectangle (pick one): One angle is a right angle The diagonals are congruent
11. 11. To prove a parallelogram is a rhombus (pick one): A pair of consecutive sides is congruent The diagonals are perpendicular One diagonal bisects a pair of opposite angles To prove that a quadrilateral is a square: It is both a rectangle and a rhombus.