Assignment

Assignment• P. 546-549: 1, 2, 3-

30 M3, 32, 34, 36, 39, 41, 44, 45

• P.733-6: 3, 12, 13, 17, 21, 24, 26, 43

• Almost a Trapezoid Worksheet

8.5: Use Properties of Trapezoids and Kites

Objectives:1. To discover and use properties of

trapezoids and kites2. To find the area of trapezoids and kites

TrapezoidsWhat makes a quadrilateral a trapezoid?

TrapezoidsA trapezoid is a

quadrilateral with exactly one pair of parallel opposite sides.

Trapezoid Parts• The parallel sides

are called bases• The non-parallel

sides are called legs

• A trapezoid has two pairs of base angles

Example 1Find the value of x.

100

xA D

B C

Trapezoid Theorem 1If a quadrilateral is a trapezoid, then the

consecutive angles between the bases are supplementary.

r

ty

xA D

B C

If ABCD is a trapezoid, then x + y = 180° and r + t = 180°.

MidsegmentA midsegment of a

trapezoid is a segment that connects the midpoints of the legs of a trapezoids.

Isosceles TrapezoidAn isosceles trapezoid is a trapezoid with

congruent legs.

Investigation 1In this Investigation, you

will be using Geometer’s Sketchpad to construct an isosceles trapezoid, and then you will discover some properties about its base angles, diagonals, and midsegment.

Trapezoid Theorem 2If a trapezoid is isosceles, then each pair of

base angles is congruent.

Trapezoid Theorem 3A trapezoid is isosceles if and only if its

diagonals are congruent.

Ti

Trapezoid Theorem 4The midsegment of a

trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

If

Example 2Find the measure of each missing angle.

Example 3For a project, you must cut an 11” by 14”

rectangular piece of poster board. Knowing how poorly you usually wield a pair of scissors, you decide to do some measuring to make sure your board is truly rectangular. Thus, you measure the diagonals and determine that they are in fact congruent. Is your board rectangular?

Example 4Find the value of x.

KitesWhat makes a quadrilateral a kite?

KitesA kite is a

quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Angles of a KiteYou can construct a kite by joining two

different isosceles triangles with a common base and then by removing that common base.

Two isosceles triangles can form one kite.

Angles of a KiteJust as in an

isosceles triangle, the angles between each pair of congruent sides are vertex angles. The other pair of angles are nonvertex angles.

Investigation 2In this Investigation, you

will be using Geometer’s Sketchpad to construct a kite. Instead of flying it, you will discover some properties about its angles and diagonals.

Kite Theorem 1If a quadrilateral is a kite, then the nonvertex

angles are congruent.

Kite Theorem 2If a quadrilateral is a kite, then the diagonal

connecting the vertex angles is the perpendicular bisector of the other diagonal.

E

A

B

C

D

and CE AE.

Kite Theorem 3If a quadrilateral is a kite, then a diagonal

bisects the opposite non-congruent vertex angles.

A

B

C

D

If ABCD is a kite, then BD bisects B and D.

Example 5Quadrilateral DEFG is

a kite. Find mD.

Example 6Find the measures of each side of kite PQRS. Write your answers in simplest radical form.

Investigation 3Now you will discover

a justification for the area formulas for trapezoids and kites.

Polygon Area Formulas

Bases and Heights 3The parallel sides of a

trapezoid are the bases. The altitude is a segment connecting the bases that is perpendicular to both. The length of the altitude (the distance between the bases) is the height.

Example 7Find the area of the trapezoid.

Example 8Find the area of each polygon.

Example 9One diagonal of a kite is twice as long as the

other diagonal. The area of the kite is 90.25 square inches. What are the lengths of the diagonals?

Assignment• P. 546-549: 1, 2, 3-

30 M3, 32, 34, 36, 39, 41, 44, 45

• P.733-6: 3, 12, 13, 17, 21, 24, 26, 43

• Almost a Trapezoid Worksheet