Use Properties of Trapezoids
and Kites
Goal: Use properties of trapezoids
and kites.
Vocabulary
Trapezoid: A trapezoid is a quadrilateral with
exactly one pair of parallel sides.
Bases of a trapezoid: The parallel sides of a
trapezoid are the bases.
Base angles of a trapezoid: A trapezoid has
two pairs of base angles. Each pair shares
a base as a side.
Vocabulary (Cont.)
Legs of a trapezoid: The nonparallel sides of
a trapezoid are the legs.
Isoceles trapezoid: An isosceles trapezoid is a
trapezoid in which the legs are congruent.
Midsegment of a trapezoid: The midsegment
of a trapezoid is the segment that connects
the midpoints of its legs.
Kite: A kite is a quadrilateral that has two pairs
of consecutive congruent sides, but opposite
sides are not congruent.
Example 1: Use a coordinate plane
Show that CDEF is a trapezoid.
Solution
Compare the slopes of opposite
sides.
Slope of DE 
4-3 1

4-1 3
2- 0
2
1
Slope of CF 


6-0
6
3
The slopes of DE and CF are the same, so DE
CF.
Example 1 (Cont.)
Slope of EF 
2-4
-2

 -1
6-4
2
3-0
3
Slope of CD 

 3
1-0
1
The slopes of EF and CD are not the same, so EF is
not
to CD.
Because quadrilateral CDEF has exactly one pair of
parallel sides, it is a trapezoid.
Theorem 6.14
If a trapezoid is
isoceles, then each
pair of base angles is
congruent.
If trapezoid ABCD is isosceles, then
A   D and  B  C.
Theorem 6.15
If a trapezoid has a pair of congruent base
angles, then it is an isosceles trapezoid.
If A  D (or if B  C ), then trapezoid ABCD is isosceles.
Theorem 6.16
A trapezoid is isosceles iff its diagonals are
congruent.
Trapezoid ABCD is isosceles iff AC  BD.
Example 2: Use properties of
isosceles trapezoids
Kitchen A shelf fitting into a cupboard in the
corner of a kitchen is an isosceles trapezoid.
Find mN , mL, and mM .
Solution
Step 1 Find mN . KLMN is an isosceles trapezoid, so
N and K are congruent base angles, and
mN  mK  50.
Example 2 (Cont.)
Step 2 Find mL. Because K and L are consecutive
interior angles formed by KL are intersecting two
parallel lines, they are supplementary. So,
mL  180  50  130.
Step 3 Find mM . Because M and L are a pair
of base angles, they are congruent, and
mM  mL  130.
So, mN  50, mL  130, and mM  130.
Checkpoint 1
1. In Example 1, suppose the coordinates of
point E are (7,5). What type of
quadrilateral is CDEF? Explain.
Parallelogram; opposite pairs of sides are
parallel.
Checkpoint 2
2. Find mC, mA, and mD in thetrapezoidshown.
mC  135, mA  45, mD  45
Theorem 6.17: Midsegment
Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base
and its length is one half the sum of the lengths of the
bases.
If MN is the midsegment of trapezoid ABCD, then
MN is parallel to AB, MN is parallel to DC,
1
and MN  ( AB  CD).
2
Example 3: Use the midsegment of
a trapezoid
In thediagram, MN is themidsegment
of trapezoidPQRS. Find MN.
Solution
Use Theorem 6.17 to find MN.
1
( PQ  SR )
2
1
 (16  9)
2
 12.5
MN 
Apply Theorem 6.17.
Substitute 16 for PQ and 9 for SR.
Simplify.
The length MN is 12.5 inches.
Checkpoint 3
3. Find MN in the trapezoid at the right.
MN = 21 ft
Theorem 6.18
If a quadrilateral is a kite, then its diagonals are
perpendicular.
If quadrilate ral ABCD is a kite,
then AC  BD.
Theorem 6.19
If a quadrilateral is a kite, then exactly one pair of
opposite angles are congruent.
If quadrilate ral ABCD is a kite and BC  BA ,
then A  C and B is not  D.
Example 4:Apply Theorem 6.19
Find mT in thekiteshown at theright.
Solution
By Theorem 6.19, QRST has exactly one
pair of congruent opposite angles. Because
Q is not  S, R and T must be  .
m R  mT . Write and solve an equation
to find mT .
Example 4 (Cont.)
mT  mR  70  88  360
Corollary to Theorem 8.1
mT  mT  70  88  360
Substitute mT for mR.
2(mT )  158  360
mT  101
Combine like terms.
Solve for mT.
Checkpoint 4
Find mG in the kite shown at the below.
mG  100