Warm Up  Elena used a rectangle, a square, a kite, a rhombus, and an isosceles trapezoid as part of a computer game she was creating. The player selects two of these shapes at random. If each of the selected shapes has at least one pair of opposite sides parallel, the player can use these shapes as keys to a higher level of the game. What is the probability of selecting a pair of keys?  Represent each shape with a letter

Given: AD is parallel to BC m< D = 8x + 20 m<A = 150 – 6x m<C = 12x + 60 Find x Find m<B Is AB parallel to DC? A C B D <A and <D are supplementary x + 8x +20 = 180 x = 5 m<D = 60 m<A = 120 m<C = 120 Since <D is supplementary to <A, AB is parallel to DC.

5.5 Properties of Quadrilaterals Objective: identify properties of quadrilaterals

Properties of parallelograms  Opposite sides are parallel and congruent  Opposite angles are congruent  Diagonals bisect each other  Any pair of consecutive angles are supplementary

Properties of rectangles:  All properties of parallelograms apply  All angles are right angles  Diagonals are congruent

Properties of a kite:  Two disjoint pairs of consecutive sides are congruent  Diagonals are perpendicular  One diagonal is the perpendicular bisector of the other  One diagonal bisects a pair of opposite angles (wy bisects <xwz and <xyz)  One pair of opposite angles are congruent (<wxy and <wzy) y W x z

Properties of a rhombus:  All properties of parallelograms apply  All properties of a kite apply  All sides are congruent (equilateral)  Diagonals bisect the angles  Diagonals are perpendicular bisectors of each other  Diagonals divide it into four congruent right triangles.

Properties of a square:  All properties of a rectangle  All properties of a rhombus  The diagonals form four isosceles triangles ( )

Properties of an isosceles trapezoid:  Legs are congruent (definition)  Bases are parallel (definition)  Lower base angles are congruent  Upper base angles are congruent  Diagonals are congruent  Lower base angle is supplementary to upper base angle

Given: ZRVA is a parallelogram AV = 2x – 4 RZ = ½ x + 8 VR = 3y + 5 ZA = y + 12 Find x Find y Find the perimeter RV AZ

R P S M O Given: Rectangle MPRS MO congruent to PO Prove:ΔROS is isosceles