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In this chapter you will learn about the special properties of quadrilaterals as well as find their perimeters and areas. You will explore the relationships of the sides and diagonals of a parallelogram, kite, trapezoid, rectangle, and rhombus. Chapter 6 – Quadrilaterals

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6.1 What If Both Sides Are Parallel? Pg. 4 Parallelograms

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6.1 – What If Both Sides are Parallel?_____ Parallelograms In the past, you used your knowledge to find the area of squares and rectangles. But what if the shape didn't have right angles?

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6.1 –PARALLELOGRAMS Find the areas of the figures below. Can you find more than one method for finding the area?

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2 4 2 8 un 2

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4 12 4 20 un 2

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6.2 –AREA OF A PARALLELOGRAM A parallelogram: a four-sided shape with two pairs of parallel sides. How can you find the area of a parallelogram? Consider this question as you answer the questions below.

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a. Keesha thinks that the rectangle and parallelograms below have the same area. Her teammate Saundra disagrees. Who is correct? Justify your conclusion.

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15 un 2

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Area of Parallelogram

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b. Does the angle at which the parallelogram slants matter? Why or why not? Explain how you know. No, the base is the same and the height is always perpendicular

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A = bh Parallelogram

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6.3 – AREA OF PARALLELOGRAMS, CONT. Several more parallelograms are shown below. In each case, find a related rectangle for which you know both the base and height. Rotating your packet might help. Use what you know about rectangles to find the area of each parallelogram.

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A = bh A = (9)(4) A = 36un 2

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A = bh A = (20)(5) A = 100un 2

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A = bh A = (7)(3) A = 21un 2

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Definition: If a quadrilateral is a parallelogram, then both pairs of ________________ sides are ______________. opposite parallel

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If a quadrilateral is a parallelogram, then both pairs of ________________ sides are ______________. opposite congruent

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If a quadrilateral is a parallelogram, then both pairs of ________________ angles are ________________. opposite congruent

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If a quadrilateral is a parallelogram, then both pairs of _______________ angles are ___________________. consecutive supplementary x y xy

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If a quadrilateral is a parallelogram, then the diagonals _______________ each other. bisect

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6.5 –PARALLELOGRAM PARTS Find the value of each variable in the parallelogram.

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a – 3 = 14 a = 17 b + 2 = 7 b = 5

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3x + 6 = 12 2y + 9 = 27 2y = 18 3x = 6 x = 2 y = 9

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130° 50°

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9b – 2 = 106 9b = 108 b = 12 7a – 3 + 106 = 180 7a + 103 = 180 a = 11 7a = 77

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If opposite sides of a quadrilateral are ________________, then the quadrilateral is a ________________. congruent parallelogram

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If both pairs of opposite angles are _________________, then the quadrilateral is a _________________. congruent parallelogram

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If consecutive angles are ________________, then the quadrilateral is a ________________. supplementary parallelogram

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If the diagonals ____________ each other, then the quadrilateral is a ________________. bisect parallelogram

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If one pair of opposite sides are ____________ and ____________, then the quadrilateral is a ________________. congruent parallelogram parallel New!!!

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6.6 –PROVING PARALLELOGRAMS Can you prove the quadrilaterals are parallelograms? Why or why not?

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yes One pair of opposite sides parallel and congruent yes Both pairs of opposite angles are congruent

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no Parallel and congruent marks are not on the same sides. yes Both pairs of opposite sides are congruent

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yes Both pairs of opposite sides are parallel no Only one pair of congruent angles

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yes Diagonals bisect each other

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6.7 –PARALLELOGRAM IDENTIFICATION The definition of a parallelogram is, "A quadrilateral with both opposite sides parallel." Based on this definition, circle all parallelograms below.

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A B C D a. What is the slopes of all four line segments? AB = ______ BC = ______ CD = ______ AD = ______ 2 7 2727 8 1 8181 2727 8181

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A B C D b. What is the relationship between these sides, given the slopes? Explain. AB = ______ BC = ______ CD = ______ AD = ______ 2 7 2727 A 8 1 8181 2727 8181 Both opposite sides are parallel

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A B C D c. What is the length of all four line segments? AB = ______ BC = ______ CD = ______ AD = ______ 2 7 A 8 1 2 2 + 7 2 = d 2 4 + 49 = d 2 53 = d 2 1 2 + 8 2 = d 2 1 + 64 = d 2 65 = d 2

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A B C D d. What is the relationship between these sides, given their length? Explain. AB = ______ BC = ______ CD = ______ AD = ______ 2 7 A 8 1 Both opposite sides are congruent

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A B C D e. What kind of quadrilateral is this? How do you know? 2 7 A 8 1 Parallelogram, Both opposite sides are parallel and congruent

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Parallelogram Rectangle Rhombus Square Trapezoid Isosceles Trapezoid Kite Triangle

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Name Block #

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Parallelogram

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Both opposite sides parallel Both opposite sides congruent Both opposite angles congruent Consecutive angles supplementary Diagonals bisect each other

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Triangle

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3 sides

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