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

Properties, Theorems, and Conclusions

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

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

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

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 NOTE: Opposite angles are already congruent!

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

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

Obviously Difficult, Secretly Simple.

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

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

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

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

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

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

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

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

Properties, Theorems, and Conclusions

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

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

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

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

Is it better then a Rhombus?

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

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

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

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

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

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

Properties, Theorems, and Conclusions

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

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

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

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

How hard can this be?

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

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

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

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

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

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