Section 6.6Trapezoids and Kites

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs. The base angles are formed by the base and one of the legs. In trapezoid ABCD, ÐA and ÐB are one pair of base angles and ÐC and ÐD are the other pair. If the legs of a trapezoid are congruent, then it is an isosceles trapezoid.

Concept 1

Example 1: Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet,

a) find mMJK.

because….

mÐJML + mÐMJK = 180 because….

130 + mÐMJK = 180 because….

mÐMJK = ______ because….

||JK LM JKLM is a trapezoid

Consec. Int. Angles Theorem

substitution

Subtract 130 from each side50°

Example 1: Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and JL = 10.3 feet,

b) find MN.

because….

JL = KM because…. JL = KN + MN because….

10.3 = 6.7 + MN because….

JL KM JKLM is an isosceles trapezoid

Definition of congruent segments

Substitution

3.6 = MN Subtract 6.7 from each side

Segment Addition

Example 2: Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

1 1 2 1Slope of

3 5 8 4AB

4 3 1

Slope of 2 2 4

CD

1 4 3Slope of 1

5 2 3AD

3 1 4Slope of 4

2 3 1AD

Exactly one pair of opposite

sides are parallel, and .

So, is a trapezoid.

AB CD

ABCD

2 22 5 4 1

18

AD

2 22 3 3 1

17

BC

Since the legs are not congruent, ABCD is not an isosceles trapezoid.

The midsegment of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid.

Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x?

1 Trapezoid Midsegment Theorem

2MN KF JG

130 20 Substitution

2x

60 20 Multiply each side by 2.x

40 Subtract 20 from each side.x

A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. Unlike a parallelogram, the opposite sides of a kite are not congruent or parallel.

Example 4: a) If WXYZ is a kite, find mXYZ.

WXY WZY because …

mWZY = ________

mW + mX + mY + mZ = _____ because …

a kite has one pair of angles which are between the two non-congruent sides.

121° substitution

360° polygon int. angles sum theorem

73° + 121° + mY + 121° = 360° Substitution

mY = 45° Simplify

Example 4: b) If MNPQ is a kite, find NP.

(NR)2 + (MR)2 = (MN)2 because…Pythagorean Theorem

(6)2 + (8)2 = MN2 Substitution

36 + 64 = MN2 Simplify.

100 = MN2 Add.

10 = MN Take the square root of each side.

MN = NP Consecutive sides of a kite are congruent.

10 = NP Substitution

Example 4: c) If BCDE is a kite, find mCDE.

C E, and the sum of the interior angles of a kite is 360°, so to find the measure of D = 360 – 130 – 130 – 64.

mD = 36°