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8.5 Trapezoids and Kites Goal 1 Using Properties of Trapezoids Goal 2 Using Properties of Kites Using Properties of Trapezoids A Trapezoid is a quadrilateral with exactly one pair of parallel sides. Trapezoid Terminology • The parallel sides are called BASES. • The nonparallel sides are called LEGS. • There are two pairs of base angles, the two touching the top base, and the two touching the bottom base. Using Properties of Trapezoids ISOSCELES TRAPEZOID - If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. Theorem - Both pairs of base angles of an isosceles trapezoid are congruent. Theorem – If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. Theorem - The diagonals of an isosceles trapezoid are congruent. Using Properties of Trapezoids Example 1 CDEF is an isosceles trapezoid with leg CD = 10 and mE = 95°. Find EF, mC, mD, and mF. Using Properties of Trapezoids When working with a trapezoid, the height may be measured anywhere between the two bases. Also, beware of "extra" information. The 35 and 28 are not needed to compute this area. 1 Area of trapezoid = hb1 b2 2 Find the area of this trapezoid. A = ½ * 26 * (20 + 42) A = 806 Using Properties of Trapezoids Example 2 Find the area of a trapezoid with bases of 10 in and 14 in, and a height of 5 in. Using Properties of Trapezoids Midsegment of a Trapezoid – segment that connects the midpoints of the legs of the trapezoid. A B Midsegment E F D C Using Properties of Trapezoids Theorem: Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases. A B EF || AB ; EF || DC 1 EF ( AB DC ) 2 Midsegment E F D C Using Properties of Trapezoids Example 4 102° A Find AB, mA, and mC E 17 in B F 65° D 24 in. C Using Properties of Trapezoids Example 5 Using Properties of Kites Using Properties of Kites A quadrilateral is a kite if and only if it has two distinct pair of consecutive sides congruent. • The vertices shared by the congruent sides are ends. • The line containing the ends of a kite is a symmetry line for a kite. • The symmetry line for a kite bisects the angles at the ends of the kite. • The symmetry diagonal of a kite is a perpendicular bisector of the other diagonal. Using Properties of Kites Theorem If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. mB = mC A C B D Using Properties of Kites Area Kite = one-half product of diagonals 1 A d1d 2 2 D 1 Area AC BD 2 A B C Using Properties of Kites Example 6 ABCD is a Kite. A a) Find the lengths of all the sides. D 2 E 4 4 4 b) Find the area of the Kite. C B Using Properties of Kites Example 7 C CBDE is a Kite. Find AC. 29 E 5 A B D Using Properties of Kites Example 8 A ABCD is a kite. Find the mA, mC, mD (x + 30)° B C 125° x° D Area of a Rectangle: The area of a rectangle is the product of its base and height. A = bh Area of a Parallelogram: The area of a parallelogram is the product of a base and its corresponding height. A = bh Area of a Triangle: The area of a triangle is one half the product of a base and its corresponding height. A = ½ bh Area of a Trapezoid: The area of a trapezoid is one half the product of the height and the sum of the bases. A = ½ h(b1 + b2) Area of a Kite: The area of a kite is one half the product of the length of its diagonals. A = ½ d1 d 2 Area of a Rhombus: The area of a rhombus is equal to one half the product of the lengths of the diagonals. A = ½ d1 d 2 Homework/Classwork