5.1 1 Properties of Trapezoids and Kites Example 1Use a coordinate plane Use a coordinate plane Show that CDEF is a trapezoid. 0 , 0 C 3 , 1 D 4 , 4 E 2 , 6 F Soluti on Compare the slopes of the opposite sides. DE of slope 3 4 1 4 3 1 CF of slope 0 2 0 6 6 2 3 1 The slopes of DE and CF are the same, so DE ___ CF. EF of slope 4 2 4 6 2 2 1 CD of m 0 3 0 1 1 3 3 The slopes of EF and CD are not the same, so EF is ______________ to CD. not parallel Because quadrilateral CDEF has exactly one pair of _______________, it is a trapezoid. parallel sides
5.11. Properties of Trapezoids and Kites. The slopes of DE and CF are the same, so DE ___ CF. The slopes of EF and CD are not the same, so EF is ______________ to CD. Use a coordinate plane. Example 1. Show that CDEF is a trapezoid. Solution. Compare the slopes of the opposite sides. - PowerPoint PPT Presentation
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5.11 Properties of Trapezoids and Kites
Example 1 Use a coordinate planeUse a coordinate plane
Show that CDEF is a trapezoid.
0 ,0C
3 ,1D 4 ,4E
2 ,6F
SolutionCompare the slopes of the opposite sides.
DE of slope 34 14 3
1
CF of slope 02 06 6
2
3
1
The slopes of DE and CF are the same, so DE ___ CF.
EF of slope 42 46 2
21
CD of m 03
01 1
3 3
The slopes of EF and CD are not the same, so EF is ______________ to CD.not parallelBecause quadrilateral CDEF has exactly one pair of _______________, it is a trapezoid.parallel sides
5.11 Properties of Trapezoids and Kites
Theorem 5.29If a trapezoid is isosceles, then each pair of
base angles is _____________.congruent
A D
B C
C. ___ and ___ A If trapezoid ABCD is isosceles, then
D B
5.11 Properties of Trapezoids and Kites
Theorem 5.30If a trapezoid has a pair of congruent
___________, then it is an isosceles trapezoid.base angles
A D
B C
,C B ifor D A If then trapezoid ABCD is isosceles.
5.11 Properties of Trapezoids and Kites
Theorem 5.31A trapezoid is isosceles if and only if its diagonals
are __________.congruent
A D
B C
____. ____ Trapezoid ABCD is isosceles if and only if
AC BD
5.11 Properties of Trapezoids and Kites
Example 2 Use properties of isosceles trapezoidsUse 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.
L
K
M
N
o50
SolutionStep 1 Find m N. KLMN is an ___________________, so N
and ___ are congruent base angles, and
isosceles trapezoid
K.______N mm K o50
Step 2
supplementary.______M mm L o130
Find m L. Because K and L are consecutive interior angles formed by KL intersecting two parallel lines, they are _________________.
5.11 Properties of Trapezoids and Kites
Example 2 Use properties of isosceles trapezoidsUse 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.
L
K
M
N
o50
SolutionStep 3 Find m M. Because M and ___ are a pair of base
angles, they are congruent, and L
.______M mm L o130.____M ____,L ____,N So, mmm o50 o130 o130
5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. In Example 1, suppose the coordinates
of E are (7, 5). What type of quadrilateral is CDEF? Explain.
0 ,0C
3 ,1D
5 ,7E
2 ,6F
DE of slope 35 17 6
2
CF of slope 02 06 6
2
3
13
1
The slopes of DE and CF are the same, so DE ___ CF.
EF of slope 52 76 1
3
3
CD of m 03 01 1
3 3
The slopes of EF and CD are the same, so EF ___ CD.
Because the slopes of DE and CD are not the opposite reciprocals of each other, they are not perpendicular.
Therefore the opposite sides are parallel and CDEF is a parallelogram.
5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. Find m C, m A, and m D
in the trapezoid shown. A
B
D
C
o135
ABCD is an isosceles trapezoid, so base angles are congruent and consecutive interior angles are supplementary.
o135C m oo 135180Am o45
o45D m
5.11 Properties of Trapezoids and Kites
Theorem 5.32 Midsegment Theorem of TrapezoidsThe midsegment of a trapezoid is parallel to each
base and its length is one half the sum of the lengths of the bases.
.________ MN and ____,MN ____,MN If MN is the midsegment of trapezoid ABCD,
A
D
B
C
M N
AB CD 2
1AB CD
5.11 Properties of Trapezoids and Kites
Example 3 Use the midsegment of a trapezoidsUse the midsegment of a trapezoids
Solution
In the diagram, MN is the midsegment of trapezoid PQRS. Find MN
P
S
Q
R
M N
Use Theorem 5.32 to find MN.
______MN ____ 2
1PQ SR
__________ 2
116 9
____12.5
Apply Theorem 5.32Apply Theorem 5.32
Substitute ___ for PQ and Substitute ___ for PQ and ___ for SR.___ for SR.
169
Simplify.Simplify.
inches. _____ is MN 12.5
16 in.
9 in.
5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2. Find MN in the trapezoid
at the right.P
S
Q
R
M
N
30 ft
12 ft
______MN ____ 2
1PS QR
__________ 2
130 12
____ 21
ft 21 is MN
5.11 Properties of Trapezoids and Kites
Theorem 5.33If a quadrilateral is a kite, then its diagonals are
_______________.
______then If quadrilateral ABCD is a kite,
AC BD
A
DB
C
perpendicular
5.11 Properties of Trapezoids and Kites
Theorem 5.34If a quadrilateral is a kite, then exactly one pair
of opposite angles are congruent.
D.B___ and CA___then
A
DB
C
If quadrilateral ABCD is a kite and BC BA,
5.11 Properties of Trapezoids and Kites
Example 4 Use properties of kitesUse properties of kites
SolutionBy Theorem 5.34, QRST has exactly one pair of __________ opposite angles.
Find m T in the kite shown at the right.
RQ
S
T
o70
o88
T. __ So, congruent. bemust T and __ ,SQ Because mmcongruent
R RT. find oequation tan solve and Write m
Corollary to Corollary to Theorem 5.16Theorem 5.16__________RT mm o70 o88 o360
__________TT mm o70 o88 o360 Substitute.Substitute.
_______T__ m Combine like terms.Combine like terms.2 o158 o360Solve.Solve.____T m o101
5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. Find m G in the kite