6.4 Rhombuses, Rectangles, and Squares

Review Find the value of the variables. p + 50° + (2p – 14)° = 180° 52° (2p-14)° 50° 68° p + 50° + (2p – 14)° = 180° p + 2p + 50° - 14° = 180° 3p + 36° = 180° 3p = 144 ° p = 48 ° 52° + 68° + h = 180° 120° + h = 180 ° h = 60°

Special Parallelograms Rhombus A rhombus is a parallelogram with four congruent sides.

Special Parallelograms Rectangle A rectangle is a parallelogram with four right angles.

Special Parallelogram Square A square is a parallelogram with four congruent sides and four right angles.

Corollaries Rhombus corollary Rectangle corollary Square corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle corollary A quadrilateral is a rectangle if and only if it has four right angles. Square corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Example PQRS is a rhombus. What is the value of b? 2b + 3 = 5b – 6

Review In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___ 1 2 3 4 5 7f – 3 = 4f + 9 3f – 3 = 9 3f = 12 f = 4

Example PQRS is a rhombus. What is the value of b? 3b + 12 = 5b – 6

Theorems for rhombus A parallelogram is a rhombus if and only if its diagonals are perpendicular. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. L

Theorem of rectangle A parallelogram is a rectangle if and only if its diagonals are congruent. A B D C

Match the properties of a quadrilateral The diagonals are congruent Both pairs of opposite sides are congruent Both pairs of opposite sides are parallel All angles are congruent All sides are congruent Diagonals bisect the angles Parallelogram Rectangle Rhombus Square B,D A,B,C,D A,B,C,D B,D C,D C

Decide if the statement is sometimes, always, or never true. A rhombus is equilateral. 2. The diagonals of a rectangle are _|_. 3. The opposite angles of a rhombus are supplementary. 4. A square is a rectangle. 5. The diagonals of a rectangle bisect each other. 6. The consecutive angles of a square are supplementary. Always Sometimes Sometimes Always Always Always Quadrilateral ABCD is Rhombus. 7. If m <BAE = 32o, find m<ECD. 8. If m<EDC = 43o, find m<CBA. 9. If m<EAB = 57o, find m<ADC. 10. If m<BEC = (3x -15)o, solve for x. 11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x 12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x. A B E D C 32o 86o 66o 35o 16 26

Coordinate Proofs Using the Properties of Rhombuses, Rectangles and Squares Using the distance formula and slope, how can we determine the specific shape of a parallelogram? Rhombus – Rectangle – Square - 1. Show all sides are equal distance 2. Show all diagonals are perpendicular. 1. Show diagonals are equal distance 2. Show opposite sides are perpendicular Show one of the above four ways. Based on the following Coordinate values, determine if each parallelogram is a rhombus, a rectangle, or square. P (-2, 3) P(-4, 0) Q(-2, -4) Q(3, 7) R(2, -4) R(6, 4) S(2, 3) S(-1, -3) RECTANGLE RECTANGLE