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3 Special Figures: The Rhombus, The Rectangle, The Square A Retrospect

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Properties, Theorems, and Conclusions

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

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All Properties of Parallelograms Work! 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

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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

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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 1 1 1 2 2 2 2 NOTE: Opposite angles are already congruent!

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

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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

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Obviously Difficult, Secretly Simple.

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Step #1: Must first show the quadrilateral is a Parallelogram! Use one of the methods for 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

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

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

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

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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 4 congruent sides Rhombus Quadrilateral Parallelogram Diagonals Perpendicular Rhombus Quadrilateral Parallelogram Diagonals bisect each pair of opposite angles Rhombus 1 1 1 1 2 2 2 2

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

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

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If you did things right, you should have only used 1 sheet of paper, right?

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Properties, Theorems, and Conclusions

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

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All Properties of Parallelograms Work! 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

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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

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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

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Is it better then a Rhombus?

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Step #1: Must first show the quadrilateral is a Parallelogram! Use one of the methods for 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

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

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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 All 4 angles congruent (all 4 angles are right angles) Rectangle

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

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

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If you did things right, you should have only used 1 sheet of paper, right?

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Properties, Theorems, and Conclusions

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

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

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All Properties of a Rhombus Work! All Properties of a Rectangle Work! Diagonals are perpendicular Diagonals bisect each pair of opposite angles Diagonals are congruent 11 11 11 11

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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

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How hard can this be?

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Step #1: Must first show the quadrilateral is a Parallelogram! Use one of the methods for 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

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

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

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

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

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If you did things right, you should have only used 1 sheet of paper, right?

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