Lesson 6.6 Trapezoids and Kites

Definition  Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Parts of a Kite Vertex angles: <B & <D Non-Vertex diagonal CA Non-Vertex angles: <A & <C Vertex diagonal BD

Properties of Kites  The diagonals of a kite are perpendicular  Non-vertex angles (a)are congruent  Vertex diagonal bisects the vertex angles  Vertex diagonal bisects the non-vertex diagonal

Perpendicular Diagonals of a Kite  If a quadrilateral is a kite, then its diagonals are perpendicular.

Non-Vertex Angles of a Kite  If a quadrilateral is a kite, then non- vertex angles are congruent  A   C,  B   D

Vertex diagonals bisect vertex angles If a quadrilateral is a kite then the vertex diagonal bisects the vertex angles.

Vertex diagonal bisects the non-vertex diagonal If a quadrilateral is a kite then the vertex diagonal bisects the non-vertex diagonal

Definition-a quadrilateral with exactly one pair of parallel sides. Leg Leg Base Base A B CD › › Trapezoid

<A + <C = 180 <A + <C = 180 <B + <D = 180 <B + <D = 180 A B CD › › Leg Angles are Supplementary Property of a Trapezoid

Isosceles Trapezoid Definition - A trapezoid with congruent legs.

Isosceles Trapezoid - Properties | | 1) Base Angles Are Congruent 2) Diagonals Are Congruent

Example PQRS is an isosceles trapezoid. Find m P, m Q and mR. m  R = 50  since base angles are congruent m  P = 130  and m  Q = 130  (consecutive angles of parallel lines cut by a transversal are  )

Find the measures of the angles in trapezoid 48 m< A = 132 m< B = 132 m< D = 48

Find BE  AC = 17.5, AE = 9.6 E

Example  Find the side lengths of the kite.

Example Continued We can use the Pythagorean Theorem to find the side lengths = (WX) = (WX) = (WX) = (XY) = (XY) = (XY) 2

Find the lengths of the sides of the kite W X Y Z

Find the lengths of the sides of kite to the nearest tenth

Example 3  Find mG and mJ. Since GHJK is a kite  G   J So 2(m  G)  + 60  = 360  2(m  G) =168  m  G = 84  and m  J = 84 

Try This!  RSTU is a kite. Find mR, mS and mT. x x = 360 2x = 360 2x = 80 x = 40 So m  R = 70 , m  T = 40  and m  S = 125 