Geometry B Bellwork

1) State whether the quadrilateral is a parallelogram. Explain your reasoning.

6.5 Trapezoids and Kites

Geometry

Using properties of trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid.

base

base

legleg

A B

D C

Trapezoid Theorems

Theorem 6-15 The base angles of

an isosceles trapezoid are congruent.

A ≅ B, C ≅ D

A B

D C

Trapezoid Theorems

Theorem 6-16 The diagonals of an

isosceles trapezoid are congruent.

ABCD is isosceles, AC BD.≅

A B

D C

Geometry B Bellwork

2) State the definition of a trapezoid. Label its parts.

Kite theorems

Theorem 6-17 The diagonals of a

kite are perpendicular

AC BD

B

C

A

D

EX. Given an isosceles trapezoid.Find the measure of each angle…

1)

77° 1

23

OYO… Given an isosceles trapezoid find the measures of the missing angles

2)

1 49 °

2 3

EX. Given a kite, find the measures of the numbered angles

45°

2

1 3

65º

OYO…Given a kite, find the measures of the numbered angles

52°

21 3

65º

4

Solve for x.

3)

4)

60° (3x + 15)°

(x + 6)°

2x°

Ex. 2: Using properties of trapezoids

Show that ABCD is a trapezoid. Compare the slopes of opposite sides.

The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5

The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3

The slopes of AB and CD are equal, so AB ║ CD.

The slope of BC = 7 – 5 = 2 = 1 4 – 0 4 2

The slope of AD = 4 – 0 = 4 = 2 7 – 5 2

The slopes of BC and AD are not equal, so BC is not parallel to AD.

So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.

8

6

4

2

5 10 15A(5, 0)

D(7, 4)

C(4, 7)

B(0, 5)

Midsegment of a trapezoid

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

midsegment

B C

DA

Theorem 6.17: Midsegment of a trapezoid

The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.

MN║AD, MN║BC MN = ½ (AD + BC)

NM

A D

CB

EX. Midsegment of a trapezoid

Find the midsegment

AD= 28, BC=12 MN = ½ (AD + BC) NM

A D

CB

Ex. 3: Finding Midsegment lengths of trapezoids

LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

Ex. 3: Finding Midsegment lengths of trapezoids

Use the midsegment theorem for trapezoids.

DG = ½(EF + CH)=½ (8 + 20) = 14” C

D

E

D

G

F

Ex. 5: Angles of a kite

Find mG and mJ

in the diagram at the

right.

SOLUTION:

GHJK is a kite, so G ≅ J and mG = mJ.2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is

360°

2(mG) = 168°Simplify

mG = 84° Divide each side by 2.

So, mJ = mG = 84°

J

G

H K132° 60°

Reminder:

Quiz after this section