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6.2 What Are Special Parallelograms? Pg. 9 Properties of Rhombi, Rectangles, and Squares 6.2 – What Are Special Parallelograms?___ Properties of Rhombi, Rectangles, and Squares In the previous lesson, you learned that parallelograms have both pairs of opposite sides parallel. You also discovered many different properties of parallelograms. Today you are going to continue your investigation with parallelograms with even more special properties. 6.8–PARALLELOGRAMS WITH RIGHT ANGLES a. Rectangles are special parallelograms. Since they are parallelograms, what do you already know about rectangles? opposite parallel Both _____________ sides are___________ Both _____________ opposite sides are congruent ________________ opposite angles are Both _____________ congruent ________________ consecutive angles are Both _____________ supplementary ________________ bisect The diagonals ________________ each other b. Mark wanted to learn more about this shape. He noticed that the diagonals seem to have a special relationship beyond just being bisected. He decided to investigate. He drew a rectangle twice, adding one diagonal. Find the length of AC and BD. Show all work. What do you notice? 2 8 + 2 15 2 x 2 x = 289 = 17 = x 2 8 + 2 15 2 x 2 x = 289 = 17 = x Diagonals are congruent c. List the two special properties Rectangles have that general Parallelograms don’t have. 4 right angles Diagonals are congruent 6.9–PARALLELOGRAMS WITH EQUAL SIDES a. A rhombus is another type of special parallelogram. Since they are parallelograms, what do you already know about rhombuses? opposite Both _____________ parallel sides are ________________ Both _____________ opposite sides are congruent ________________ opposite angles are Both _____________ congruent ________________ consecutive angles are Both _____________ supplementary ________________ x y 180 y x y x bisect The diagonals ________________ each other c. Audrey wanted to learn more about her shape. She noticed that the diagonals seem to have a special relationship as well. She measured the sides of the rhombus and all were 5 units long. Then she measured AC = 6 units and BD = 8 units. Mark these lengths on the picture below. Is there a way to tell if ∆AEB is a right triangle? Explain. 2 5 5 5 3 4 3 5 4 5 2 3 2 4 = + 25 = 9 + 16 25 = 25 The diagonals are perpendicular d. Audrey noticed something else with the angle in the rhombus. Using the given lines symmetry, mark any angles congruent. What do you notice? Diagonals bisect the angles c. List the two special properties Rhombuses have that general Parallelograms don’t have. 4 congruent sides Diagonals are perpendicular Diagonals bisect angles 6.10 – PARALLELOGRAMS WITH EQUAL SIDES AND RIGHT ANGLES Ms. Matthews has a favorite quadrilateral. It is a rhombus combined with a rectangle. a. What is the name of Ms. Matthews' shape? Draw a picture to support your answer. square b. This shape has more properties than any other quadrilateral. Why do you think this is? It is a parallelogram, a rectangle, and a rhombus 6.11 – SPECIAL PARALLELOGRAMS Name the type of parallelogram. Explain how you know using only the markings. parallelogram rectangle rhombus rhombus rectangle rhombus square rhombus 6.12 – MISSING PARTS Find the missing information based on the type of shape and its special properties. a. The diagonals of rhombus PQRS intersect at T. Find the indicated measure. mQPR _____ 30° 90° mQTP _________ mPQT _________ 60° 12 RP = _________ SP = _________ 15 15 RS = _________ 15 60° 90° 30° 15 15 b. The diagonals of rectangle WXYZ intersect at P. Given that XZ = 12, find the indicated measure. 40° WXZ _________ 40° PYX _________ 50° 80° 50° XPY _________ 80° 6 WP = _________ c. The diagonals of square DEFG intersect at H. Given that EH = 5, find the indicated measure. GHF 90° HGF 45° 45° HFG HF = 5 90° 45° 45° 6.13 – AREA Find the area of the rhombus by finding the area of each triangle and then adding. 22 275 275 25 275 275 A = 1100 ft2 1.5 7 42 = x2 + 32 16 = x2 + 9 7 1.5 7 1.5 7 7 3 1.5 7 7 = x2 7x A 6 7cm 2 8 4 8 4 84 8 A = 32 m2 4.5 3 3 3 3 4.5 3 4.5 3 3 3 4.5 3 3 6 A 18 3 ft 2 Parallelogram Trapezoid Rectangle Isosceles Trapezoid Rhombus Kite Square Triangle Rectangle • All the properties of a parallelogram • 4 right angles • Diagonals are congruent A bh Rhombus • All the properties of a parallelogram • Diagonals are perpendicular • Diagonals bisect angles Add area of each triangle Square • All the properties listed above A s 2 or A bh