6-6 Trapezoids & Kites The student will be able to: Recognize and apply the properties of trapezoids. Recognize and apply the properties of kites.

Trapezoids Trapezoid – a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs. If the legs of a trapezoid are congruent, then it is an isosceles trapezoid. An isosceles trapezoid has the following properties: The legs are congruent. Each pair of base angles is congruent. The diagonals are congruent.

What do we know about base angles of an isosceles trapezoid? Example 1: The speaker shown is an isosceles trapezoid. If inches, and inches, find each measure. a. 8 What do we know about base angles of an isosceles trapezoid? 19 The base angles are congruent, so 85 85 What do we know about consecutive interior angles? They are supplementary. 85 + x = 180 -85 x = 95

Example 1: The speaker shown is an isosceles trapezoid Example 1: The speaker shown is an isosceles trapezoid. If inches, and inches, find each measure. b. What do we know about diagonals of an isosceles trapezoid? The diagonals are congruent, so and 8 + x = 19 - 8 x = 11

Midsegment of a trapezoid: The segment that joins midpoints of the legs of a trapezoid. The midsegment has the following properties: it is parallel to each base it bisects each leg its measure is ½ the sum of the lengths of the bases = ½( )

Example 2: In the figure, is the midsegment of trapezoid FGJK. What is the value of x? = ½( ) 30 = ½( 20 + x) 30 = 10 + ½x -10 20 = ½x 40 = x

Trapezoid ABCD is shown below. If is parallel to . What is the x-coordinate of point G? Point G is the midpoint of so use the midpoint formula to find G. m = (10.5 , 2) Are we through? No. The x-coordinate of G is 10.5.

Kite – a quadrilateral with exactly 2 pairs of consecutive congruent sides. A kite has the following properties: Diagonals are perpendicular. One pair of opposite angles is congruent. It’s always the angle between two sides that aren’t congruent.

What do we know about angles of a kite? Example 4: If WXYZ is a kite, find What do we know about angles of a kite? 121 The angles between the two non-congruent sides are =. What is the sum of the interior angles of a kite? 360 73 + 121 + 121 + x = 360 315 + x = 360 -315 x = 45

Example 5: If MNPQ is a kite, find What do we know about angles formed by the diagonals of a kite? They are right angles. What do we use to find the lengths of the segments of right triangles? The Pythagorean Theorem a2 + b2 = c2 82 + 62 = c2 64 + 36 = c2 100 = c2 10 = c

Example 5: The vertices of ABCD are A(-3, -1), B(-1,3) C(2, 3), and D(4, -1). Verify that ABCD is a trapezoid. Is it an isosceles trapezoid? What will tell us it’s a trapezoid? What will tell us it’s a trapezoid? 1 pair of parallel bases What will tell us it’s an isosceles trapezoid? Congruent legs What formulas can we use to find if the bases are parallel & legs congruent? Slope 4 times and distance formula for only the legs.

The vertices of ABCD are A(-3, -1), B(-1,3) C(2, 3), and D(4, -1). Verify that ABCD is a trapezoid. Is it an isosceles trapezoid? Is it a trapezoid? yes Is it an isosceles trapezoid? Yes. There are exactly 1 pair of parallel lines and the legs are congruent.