Lesson 6 – 6 Trapezoids and Kites Geometry Lesson 6 – 6 Trapezoids and Kites Objective: Apply properties of trapezoids. Apply properties of kites.

Trapezoid What is a trapezoid? A quadrilateral with exactly one pair of parallel sides. Bases – the parallel sides Legs – the nonparallel sides Base angles – the angles formed by the base and one of the legs Isosceles trapezoid congruent legs

Theorem Theorem 6.21 If a trapezoid is isosceles, then each pair of base angles is congruent.

Theorem Theorem 6.22 If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

Theorem Theorem 6.23 A trapezoid is isosceles if and only if its diagonals are congruent.

The speaker shown is an isosceles trapezoid. If m FJH = 85, FK = 8 in The speaker shown is an isosceles trapezoid. If m FJH = 85, FK = 8 in. and JG = 19 in. Find each measure. KH 95 8 in 19 95 85 85 8 + KH = 19 FH = JG KH = 11 FH = 19 FK + KH = 19

WXYZ is Isosceles trap. 10 cm 15 cm 45 XZ XV 45 135 25 cm 10 cm

To be a trapezoid the quad must have one set of parallel sides. Quadrilateral ABCD has vertices A (-3, 4) B (2, 5) C (3, 3) and D (-1, 0). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. To be a trapezoid the quad must have one set of parallel sides. Slope of AD = -2 Slope of BC = -2 The figure is a trapezoid (AB and DC obviously not parallel) To be isosceles the diagonals must be congruent. Trapezoid ABCD is not an isosceles trapezoid.

Quadrilateral QRST has vertices Q(8, -4) R(0, 8) S(6, 8) T (-6, -10). Determine whether it is an isosceles trapezoid. Slope of QR = 3/2 Slope of RS = 0 Slope of QT = 3/7 Slope of TS = 3/2 Since one pair of parallel sides QRST is a trapezoid. Trapezoid QRST is not an isosceles trapezoid.

Midsegment of a Trapezoid The segment that connects the midpoints of the legs of the trapezoid.

Theorem Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases.

In the figure segment LH is the midsegment of trapezoid FGJK In the figure segment LH is the midsegment of trapezoid FGJK. What is the value of x? (2) (2) 30 = x + 18.2 11.8 = x

G is the midpoint of segment DC. The x-coordinate is 10.5 Trapezoid ABCD is shown below. If FG II AD, what is the x-coordinate of point G? G is the midpoint of segment DC. The x-coordinate is 10.5

Kite A quadrilateral with exactly two pairs of consecutive congruent sides.

Theorem Theorem 6.25 If a quadrilateral is a kite, then its diagonals are perpendicular. Where else have learned about the diagonals being perpendicular? Square & Rhombus Why does this work for all 3 figures? Is a Square and a Rhombus considered a Kite?

Theorem Theorem 6.26 If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.

pair of opposite congruent angles x x If FGHJ is a kite, find the measure of angle GFJ Kites have exactly one pair of opposite congruent angles x x 2x + 128 + 72 = 360 2x + 200 = 360 2x = 160 x = 80

If WXYZ is a kite, find ZY. (PZ)2 + (PY)2 = (ZY)2 82 + 242 = (ZY)2 Remember to simplify!

x 50 x 38 2x + 38 + 50 = 360 2x + 88 = 360 2x = 272 x = 136

5 8 If BT = 5 and TC = 8, find CD. (BT)2 + (TC)2 = (BC)2 BC = CD

Homework Pg. 440 1 – 7 all, 8 – 24 EOE, 36 – 48 EOE, 66, 70 – 76 E, 82