6.6 Trapezoids and Kites Check.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids). Spi.3.2 Use coordinate geometry to prove characteristics of polygonal figures. Check.4.14 Identify and use medians, midsegments, altitudes, angle bisectors, and perpendicular bisectors of triangles to solve problems (e.g., find segment lengths, angle measures, points of concurrency).

Trapezoids A trapezoid is a quadrilateral with only 1 pair of parallel sides. The median of a trapezoid is parallel to the bases and its measure its ½ the sum of the measure of the bases EF = ½(AB + DC) "Even if you're on the right track, you'll get run over if you just sit there." Will Rogers A D C B base leg median E F The diagonals of an isosceles trapezoid are congruent. Both Pairs of Base Angles are congruent.  A   B and  D   C G H I J If GI  HJ then GJ  HI PREPARE FOR CONSTRUCTIONS

Constructions Draw a segment AD. Place the compass point at A, open to the width of AD and draw an arc above AD. Label any point on the arc as B. Using the same setting, place the compass at B and drawn an arc to the right of B. Place the compass at D, draw an arc to intersect the arc drawn from B. Label as C. Use the straight edge to complete. A D B C

Constructions - Kite Draw a segment RT. Choose a compass setting greater than ½ RT. Place compass at R and make an arc above and below the line. Increase the compass settings and repeat with same setting at point T. Find the points where the arcs intersect and label as Q and S. Draw QRST R T Q S What special properties do you notice about this quadrilateral? Symmetric, Sides and Angles, bisectors perpendicular

Use Properties of Kites A. If WXYZ is a kite, find m  XYZ. Since a kite only has one pair of congruent angles, which are between the two non- congruent sides,  WXY   WZY. So,  WZY = 121 . m  W + m  X + m  Y + m  Z=360Polygon Interior Angles Sum Theorem m  Y + 121=360Substitution m  Y=45Simplify. Answer: m  XYZ = 45

A.5 B.6 C.7 D.8 B. If JKLM is a kite, find KL.

Constructions – Median of Trapezoid Draw a Trapezoid WXYZ. Construct the perpendicular bisectors of XY and WZ. Label the midpoints as M and N. Draw MN M X W N Y Z Measure WX, ZY and MN What do you find?

Trapezoids A trapezoid is a quadrilateral with only 1 pair of parallel sides. Both Pairs of Base Angles are congruent.  A   B and  D   C The median of a trapezoid is parallel to the bases and its measure its ½ the sum of the measure of the bases EF = ½(AB + DC) A D C B base leg median E F The diagonals of an isosceles trapezoid are congruent. G H I J If GI  HJ then GJ  HI Objective: Understand and apply the properties of trapezoids be able to solve problems using the medians of trapezoids.

Identify a Trapezoid J(-18, -1), K(-6, 8), L(18, 1), M (-18, -26) 1. Verify that JKLM is a Trapezoid 2. Is JKLM an isosceles trapezoid? 2 Sides are parallel making it a trapezoid 2 Legs are equivalent making it an isosceles trapezoid

A(5, 1), B(-3,-1), C(-2, 3) and D(2,4) Determine if ABCD is a trapezoid Slope AB = ¼, CD = ¼ Slope AD = -1, Slope BC = 4 2 sides are parallel so it is a Trapezoid Determine if it is isosceles BC = √17 and AD= √18 No it is not an isosceles trapezoid

Medians of Trapezoid QRST is an isosceles trapezoid with median XY. Find TS if QR = 22 and XY = 15 Find m  1, m  2, m  3, m  4 if m  1=4a -10 and m  3 = 3a XY = ½ (QR + TS) 15 = ½ (22 + TS) 30 = 22 + TS TS = 8 m  1+ m  3 = 180 4a a = 180 7a – 22.5 = 180 7a = a = 22.5 m  1=4(22.5) -10= 80 m  3 = 100, m  2= 100 m  4 = 80

DEFG is an isosceles trapezoid with median MN 1.Find DG if EF = 20 DG = 40 1.Find m  1, m  2, m  3, m  4 if m  1 = 3x + 5 and m  3 = 6x – 5 m  1 = 65, m  2 = 65, m  115, m  4= 115

Summary Trapezoid is quadrilateral with 1 pair of sides parallel. Median of a trapezoid is –P–Parallel to the bases and equal to ½ sum of the bases For an isosceles trapezoid, –t–the legs and the diagonals are congruent –b–base angles of a trapezoid are congruent. Practice Assignment –P–Page Even