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Special Parallelograms

Special Parallelograms. 3 Special Figures: The Rhombus, The Rectangle, The Square A Retrospect. The “Forgotten” Rhombus. Properties, Theorems, and Conclusions. Definition of a Rhombus A parallelogram with ALL 4 sides congruent. All Properties of Parallelograms Work!.

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Special Parallelograms

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  1. Special Parallelograms 3 Special Figures: The Rhombus, The Rectangle, The Square A Retrospect

  2. The “Forgotten” Rhombus Properties, Theorems, and Conclusions

  3. Definition of a Rhombus • A parallelogram with ALL 4 sides congruent.

  4. All Properties of Parallelograms Work! Rhombi = Parallelograms! Both pairs of opposite sides parallel All Sides Congruent! Both pairs of opposite angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other

  5. Theorem #43 • A quadrilateral is a rhombus if and only if its diagonals are perpendicular. Both pairs of opposite sides parallel All Sides Congruent! Both pairs of opposite angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other Diagonals are perpendicular

  6. Theorem #44 • A quadrilateral is a rhombus if and only if its diagonals bisect each pair of opposite angles. Both pairs of opposite sides parallel All Sides Congruent! Both pairs of opposite angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other Diagonals are perpendicular Diagonals bisect each pair of opposite angles 1 2 1 2 2 1 2 1 NOTE: Opposite angles are already congruent!

  7. Example #1 • Name pairs of parallel segments. • Name pairs of congruent segments. • Name pairs of congruent angles. ANSWERS: A B E D C

  8. Rhombus “HOT FACTS” • 4 Sides – Quadrilateral • Parallelogram • 2 pairs of opposite sides parallel • ALL PAIRS of opposite sides congruent • 2 pairs of opposite angles congruent • 4 pairs of consecutive angles supplementary • Diagonals bisect each other • Diagonals perpendicular • Diagonals bisects each pair of opposite angles

  9. Proving A Quadrilateral Is A Rhombus Obviously Difficult, Secretly Simple.

  10. Step #1: Must first show the quadrilateral is a Parallelogram! • Use one of the methods for parallelograms! Parallelograms BOTH pairs of opposite sides congruent parallelogram BOTH pairs of opposite angles congruent parallelogram A pair of consecutive angles supplementary parallelogram Diagonals bisect each other parallelogram Exactly 1 pair of opposite sides congruent and parallel parallelogram

  11. Step #2: Once a parallelogram, then get specific! • 3 ways to show a parallelogram is a rhombus!

  12. Definition of a Rhombus • If a quadrilateral is a parallelogram and all 4 sides are congruent, then the quadrilateral is a rhombus. Quadrilateral Parallelogram Rhombus Quadrilateral  Parallelogram  4 congruent sides  Rhombus

  13. Theorem #45 • If a quadrilateral is a parallelogram and the diagonals are perpendicular, then the quadrilateral is a rhombus. Quadrilateral Parallelogram Rhombus Quadrilateral  Parallelogram  4 congruent sides  Rhombus Quadrilateral  Parallelogram  Diagonals Perpendicular  Rhombus

  14. Theorem #46 • If a quadrilateral is a parallelogram and the diagonals bisect each pair of opposite angles, then the quadrilateral is a rhombus. Quadrilateral Parallelogram Rhombus Quadrilateral  Parallelogram  4 congruent sides  Rhombus Quadrilateral  Parallelogram  Diagonals Perpendicular  Rhombus Quadrilateral  Parallelogram  Diagonals bisect each pair of opposite angles  Rhombus 1 2 1 2 2 1 2 1

  15. Area of a Rhombus (Method #1) • Theorem #53: Area of a Rhombus • Area = Base * Height • A = b*h h b

  16. Area of a Rhombus (Method #2) • Theorem #57: Area of a Rhombus • Area = ½ * diagonal 1 * diagonal 2 • A = ½ * d1 * d2 d1 d2

  17. Now don’t forget about a Rhombus! If you did things right, you should have only used 1 sheet of paper, right?

  18. The “Regal” Rectangle Properties, Theorems, and Conclusions

  19. Definition of a Rectangle • A parallelogram with ALL 4 angles congruent (ALL 4 angles are right angles)

  20. All Properties of Parallelograms Work! Rectangles = Parallelograms! Both pairs of opposite sides parallel 2 pairs of opposite sides congruent ALL 4 angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other

  21. Theorem #47 • A quadrilateral is a rectangle if and only if its diagonals are congruent. Both pairs of opposite sides parallel All Angles Congruent! Both pairs of opposite angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other Diagonals are congruent

  22. Rectangle “HOT FACTS” • 4 Sides – Quadrilateral • Parallelogram • 2 pairs of opposite sides parallel • 2 pairs of opposite sides congruent • ALL angles congruent (ALL angles are right angles) • 4 pairs of consecutive angles supplementary • Diagonals bisect each other • Diagonals Congruent

  23. Proving A Quadrilateral Is A Rectangle Is it better then a Rhombus?

  24. Step #1: Must first show the quadrilateral is a Parallelogram! • Use one of the methods for parallelograms! Parallelograms BOTH pairs of opposite sides congruent parallelogram BOTH pairs of opposite angles congruent parallelogram A pair of consecutive angles supplementary parallelogram Diagonals bisect each other parallelogram Exactly 1 pair of opposite sides congruent and parallel parallelogram

  25. Step #2: Once a parallelogram, then get specific! • 2 ways to show a parallelogram is a rectangle!

  26. Definition of a Rectangle • If a quadrilateral is a parallelogram and has all 4 angles congruent (or all 4 angles are right angles), then the quadrilateral is a rectangle. Quadrilateral  Parallelogram  Rectangle Quadrilateral  Parallelogram  All 4 angles congruent (all 4 angles are right angles)  Rectangle

  27. Theorem # 48 • If a quadrilateral is a parallelogram and its diagonals are congruent, then the quadrilateral is a rectangle. Quadrilateral  Parallelogram  Rectangle Quadrilateral  Parallelogram  All 4 angles congruent (all 4 angles are right angles)  Rectangle Quadrilateral  Parallelogram  Diagonals congruent  Rectangle

  28. Area of a Rectangle • Area = Length * Width or Base * Height • A = l * w or b * h w l

  29. Respect the Rectangle! If you did things right, you should have only used 1 sheet of paper, right?

  30. The “Sassy” Square Properties, Theorems, and Conclusions

  31. Definition of a Square • A parallelogram that is BOTH a Rhombus and a Rectangle! • (All 4 sides congruent) • (All 4 angles congruent)

  32. All Properties of Parallelograms Work! Square = Parallelograms Both pairs of opposite sides parallel ALL 4 sides congruent ALL 4 angles congruent Pairs of consecutive angles are supplementary Diagonals bisect each other

  33. All Properties of a Rhombus Work! • All Properties of a Rectangle Work! Squares = Parallelograms, Rhombi, and Rectangles Diagonals are perpendicular Diagonals bisect each pair of opposite angles Diagonals are congruent 1 1 1 1 1 1 1 1

  34. Square “HOT FACTS” • 4 Sides – Quadrilateral • Parallelogram • 2 pairs of opposite sides parallel • ALL sides congruent • ALL angles congruent (ALL angles are right angles) • 4 pairs of consecutive angles supplementary • Diagonals bisect each other • Rhombus • Diagonals perpendicular • Diagonals bisect each pair of opposite angles • Rectangle • Diagonals congruent

  35. Proving A Quadrilateral Is A Square How hard can this be?

  36. Step #1: Must first show the quadrilateral is a Parallelogram! • Use one of the methods for parallelograms! Parallelograms BOTH pairs of opposite sides congruent parallelogram BOTH pairs of opposite angles congruent parallelogram A pair of consecutive angles supplementary parallelogram Diagonals bisect each other parallelogram Exactly 1 pair of opposite sides congruent and parallel parallelogram

  37. Step #2: Once a parallelogram, then show it is a Rhombus! • Use one of the methods for Rhombus! Parallelograms  Rhombus Quadrilateral  Parallelogram  4 congruent sides  Rhombus Quadrilateral  Parallelogram  Diagonals Perpendicular Quadrilateral  Parallelogram  Diagonals bisect each pair of opposite angles

  38. Step #3: Once a parallelogram and a rhombus, then show it is a rectangle! • Use one of the methods for Rectangle! Parallelograms  Rhombus  Rectangle Quadrilateral  Parallelogram  All 4 angles congruent (all 4 angles are right angles)  Rectangle Quadrilateral  Parallelogram  Diagonals congruent  Rectangle

  39. Step #4: Call your shape a square! Quadrilateral  Parallelogram  Rhombus  Rectangle  Square

  40. Area of a Square • Postulate #22 • Area = Side * Side or Side Squared • A = s * s • Theorem #53 • Area = base * height • A = b * h h s b

  41. That’s a Square, folks! If you did things right, you should have only used 1 sheet of paper, right?

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