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6.3 Rectangles

6.3 Rectangles. Properties of Special Parallelograms. In this lesson, you will study a special types of parallelograms: rectangles. A rectangle is a parallelogram with four right angles. Properties of Rectangles. Practice:. Practice:. 2x + y = 36 x – y = 9 Solving System of Equations

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6.3 Rectangles

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  1. 6.3 Rectangles

  2. Properties of Special Parallelograms • In this lesson, you will study a special types of parallelograms: rectangles A rectangle is a parallelogram with four right angles.

  3. Properties of Rectangles

  4. Practice:

  5. Practice: 2x + y = 36 x – y = 9 Solving System of Equations 3x = 45 X = 15 Y = 6

  6. Practice: x + 11 = y x + 2 = y – 3x Solving System of Equations 9 = 3x X = 3 Y = 14

  7. Practice: 4x + 8 + 5x – 8 = 90 Solving Equation 9x = 90 X = 10

  8. Practice: x = 6x - 8 Solving Equation - 5 x = -8 X = 1.6

  9. 6.4 Rhombi and Squares

  10. Properties of Special Parallelograms • In this lesson, you will study two special types of parallelograms: rhombi and squares. A square is a parallelogram with four congruent sides and four right angles. A rhombus is a parallelogram with four congruent sides

  11. Take note: • Rhombus: A quadrilateral is a rhombus if and only if it has four congruent sides. • Rectangle: A quadrilateral is a rectangle if and only if it has four right angles. • Square: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  12. Ex. 3: Using properties of a Rhombus • In the diagram at the right, PQRS is a rhombus. What is the value of y? All four sides of a rhombus are ≅, so RS = PS. 5y – 6 = 2y + 3 5y = 2y + 9 3y = 9 y = 3

  13. The following theorems are about diagonals of rhombi and rectangles. A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC BD. Using diagonals of special parallelograms

  14. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. ABCD is a rhombus if and only if AC bisects DAB and BCD and BD bisects ADC and CBA. Using diagonals of special parallelograms

  15. Find the lengths of the sides of ABCD. Use the distance formula (See – you’re never going to get rid of this) AB=√(b – 0)2 + (0 – a)2 = √b2 + a2 BC= √(0 - b)2 + (– a - 0)2 = √b2 + a2 CD= √(- b – 0)2 + [0 - (– a)]2 = √b2 + a2 DA= √[(0 – (- b)]2 + (a – 0)2 = √b2 + a2 Given: ABCD is a parallelogram, AC  BD.Prove: ABCD is a rhombus A(0, a) D(- b, 0) B(b, 0) C(0, - a) All the side lengths are equal, so ABCD is a rhombus.

  16. Practice:

  17. Practice: 4x- 3 = 18 + x 3 x – 3 = 18 3x = 21 X = 7

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