Properties of Kites

1
Properties of Kites and Trapezoids
6-6
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
2
Warm Up Solve for x. 1. x2 38 3x2 12 2.
137 x 180 3. 4. Find FE.
5 or 5
43
156
3
Objectives
Use properties of kites to solve problems. Use
properties of trapezoids to solve problems.
4
Vocabulary
kite trapezoid base of a trapezoid leg of a
trapezoid base angle of a trapezoid isosceles
trapezoid midsegment of a trapezoid
5
A kite is a quadrilateral with exactly two pairs
of congruent consecutive sides.
6
(No Transcript)
7
Example 1 Problem-Solving Application
Lucy is framing a kite with wooden dowels. She
uses two dowels that measure 18 cm, one dowel
that measures 30 cm, and two dowels that measure
27 cm. To complete the kite, she needs a dowel to
place along . She has a dowel that is 36 cm
long. About how much wood will she have left
after cutting the last dowel?
8
Example 1 Continued
The answer will be the amount of wood Lucy has
left after cutting the dowel.
9
Example 1 Continued
N bisects JM.
Pythagorean Thm.
Pythagorean Thm.
10
Example 1 Continued
Lucy needs to cut the dowel to be 32.4 cm long.
The amount of wood that will remain after the cut
is, 36 32.4 ? 3.6 cm Lucy will have 3.6 cm
of wood left over after the cut.
11
Example 1 Continued
Look Back
12
Check It Out! Example 1
What if...? Daryl is going to make a kite by
doubling all the measures in the kite. What is
the total amount of binding needed to cover the
edges of his kite? How many packages of binding
must Daryl buy?
13
Check It Out! Example 1 Continued
The answer has two parts. the total length of
binding Daryl needs the number of packages of
binding Daryl must buy
14
Check It Out! Example 1 Continued
The diagonals of a kite are perpendicular, so the
four triangles are right triangles. Use the
Pythagorean Theorem and the properties of kites
to find the unknown side lengths. Add these
lengths to find the perimeter of the kite.
15
Check It Out! Example 1 Continued
Pyth. Thm.
Pyth. Thm.
16
Check It Out! Example 1 Continued
Daryl needs approximately 191.3 inches of
binding. One package of binding contains 2 yards,
or 72 inches.
In order to have enough, Daryl must buy 3
packages of binding.
17
Check It Out! Example 1 Continued
Look Back
18
Example 2A Using Properties of Kites
In kite ABCD, m?DAB 54, and m?CDF
52. Find m?BCD.
Kite ? cons. sides ?
?BCD is isos.
2 ? sides ?isos. ?
isos. ? ?base ?s ?
?CBF ? ?CDF
Def. of ? ? s
m?CBF m?CDF
Polygon ? Sum Thm.
m?BCD m?CBF m?CDF 180
19
Example 2A Continued
m?BCD m?CBF m?CDF 180
Substitute m?CDF for m?CBF.
m?BCD m?CBF m?CDF 180
Substitute 52 for m?CBF.
m?BCD 52 52 180
Subtract 104 from both sides.
m?BCD 76
20
Example 2B Using Properties of Kites
In kite ABCD, m?DAB 54, and m?CDF
52. Find m?ABC.
Kite ? one pair opp. ?s ?
?ADC ? ?ABC
Def. of ? ?s
m?ADC m?ABC
Polygon ? Sum Thm.
m?ABC m?BCD m?ADC m?DAB 360
Substitute m?ABC for m?ADC.
m?ABC m?BCD m?ABC m?DAB 360
21
Example 2B Continued
m?ABC m?BCD m?ABC m?DAB 360
Substitute.
m?ABC 76 m?ABC 54 360
Simplify.
2m?ABC 230
m?ABC 115
Solve.
22
Example 2C Using Properties of Kites
In kite ABCD, m?DAB 54, and m?CDF
52. Find m?FDA.
Kite ? one pair opp. ?s ?
?CDA ? ?ABC
m?CDA m?ABC
Def. of ? ?s
? Add. Post.
m?CDF m?FDA m?ABC
52 m?FDA 115
Substitute.
Solve.
m?FDA 63
23
Check It Out! Example 2a
In kite PQRS, m?PQR 78, and m?TRS 59. Find
m?QRT.
Kite ? cons. sides ?
?PQR is isos.
2 ? sides ? isos. ?
?RPQ ? ?PRQ
isos. ? ? base ?s ?
Def. of ? ?s
m?QPT m?QRT
24
Check It Out! Example 2a Continued
Polygon ? Sum Thm.
m?PQR m?QRP m?QPR 180
Substitute 78 for m?PQR.
78 m?QRT m?QPT 180
Substitute.
78 m?QRT m?QRT 180
78 2m?QRT 180
Substitute.
Subtract 78 from both sides.
2m?QRT 102
m?QRT 51
Divide by 2.
25
Check It Out! Example 2b
In kite PQRS, m?PQR 78, and m?TRS 59. Find
m?QPS.
Kite ? one pair opp. ?s ?
?QPS ? ?QRS
? Add. Post.
m?QPS m?QRT m?TRS
Substitute.
m?QPS m?QRT 59
Substitute.
m?QPS 51 59
m?QPS 110
26
Check It Out! Example 2c
In kite PQRS, m?PQR 78, and m?TRS 59. Find
each m?PSR.
Polygon ? Sum Thm.
m?SPT m?TRS m?RSP 180
Def. of ? ?s
m?SPT m?TRS
Substitute.
m?TRS m?TRS m?RSP 180
Substitute.
59 59 m?RSP 180
Simplify.
m?RSP 62
27
A trapezoid is a quadrilateral with exactly one
pair of parallel sides. Each of the parallel
sides is called a base. The nonparallel sides are
called legs. Base angles of a trapezoid are two
consecutive angles whose common side is a base.
28
If the legs of a trapezoid are congruent, the
trapezoid is an isosceles trapezoid. The
following theorems state the properties of an
isosceles trapezoid.
29
(No Transcript)
30
(No Transcript)
31
Example 3A Using Properties of Isosceles
Trapezoids
Find m?A.
Same-Side Int. ?s Thm.
m?C m?B 180
Substitute 100 for m?C.
100 m?B 180
Subtract 100 from both sides.
m?B 80
Isos.? trap. ?s base ?
?A ? ?B
Def. of ? ?s
m?A m?B
Substitute 80 for m?B
m?A 80
32
Example 3B Using Properties of Isosceles
Trapezoids
KB 21.9m and MF 32.7. Find FB.
Isos. ? trap. ?s base ?
Def. of ? segs.
KJ FM
Substitute 32.7 for FM.
KJ 32.7
Seg. Add. Post.
KB BJ KJ
Substitute 21.9 for KB and 32.7 for KJ.
21.9 BJ 32.7
Subtract 21.9 from both sides.
BJ 10.8
33
Example 3B Continued
Same line.
Isos. trap. ? ?s base ?
?KFJ ? ?MJF
Isos. trap. ? legs ?
SAS
?FKJ ? ?JMF
CPCTC
?BKF ? ?BMJ
Vert. ??s ?
?FBK ? ?JBM
34
Example 3B Continued
Isos. trap. ? legs ?
AAS
?FBK ? ?JBM
CPCTC
Def. of ? segs.
FB JB
Substitute 10.8 for JB.
FB 10.8
35
Check It Out! Example 3a
Find m?F.
Same-Side Int. ?s Thm.
m?F m?E 180
Isos.? trap. ?s base ?
?E ? ?H
Def. of ? ?s
m?E m?H
Substitute 49 for m?E.
m?F 49 180
m?F 131
Simplify.
36
Check It Out! Example 3b
JN 10.6, and NL 14.8. Find KM.
Isos.? trap. ?s base ?
Def. of ? segs.
KM JL
Segment Add Postulate
JL JN NL
Substitute.
KM JN NL
Substitute and simplify.
KM 10.6 14.8 25.4
37
Example 4A Applying Conditions for Isosceles
Trapezoids
Find the value of a so that PQRS is isosceles.
Trap. with pair base ?s ? ? isosc. trap.
?S ? ?P
m?S m?P
Def. of ? ?s
Substitute 2a2 54 for m?S and a2 27 for m?P.
2a2 54 a2 27
Subtract a2 from both sides and add 54 to both
sides.
a2 81
a 9 or a 9
Find the square root of both sides.
38
Example 4B Applying Conditions for Isosceles
Trapezoids
AD 12x 11, and BC 9x 2. Find the value of
x so that ABCD is isosceles.
Diags. ? ? isosc. trap.
Def. of ? segs.
AD BC
Substitute 12x 11 for AD and 9x 2 for BC.
12x 11 9x 2
Subtract 9x from both sides and add 11 to both
sides.
3x 9
Divide both sides by 3.
x 3
39
Check It Out! Example 4
Find the value of x so that PQST is isosceles.
Trap. with pair base ?s ? ? isosc. trap.
?Q ? ?S
Def. of ? ?s
m?Q m?S
Substitute 2x2 19 for m?Q and 4x2 13 for m?S.
2x2 19 4x2 13
Subtract 2x2 and add 13 to both sides.
32 2x2
Divide by 2 and simplify.
x 4 or x 4
40
The midsegment of a trapezoid is the segment
whose endpoints are the midpoints of the legs. In
Lesson 5-1, you studied the Triangle Midsegment
Theorem. The Trapezoid Midsegment Theorem is
similar to it.
41
(No Transcript)
42
Example 5 Finding Lengths Using Midsegments
Find EF.
Trap. Midsegment Thm.
Substitute the given values.
Solve.
EF 10.75
43
Check It Out! Example 5
Find EH.
Trap. Midsegment Thm.
Substitute the given values.
Simplify.
Multiply both sides by 2.
33 25 EH
Subtract 25 from both sides.
13 EH
44
Lesson Quiz Part I
1. Erin is making a kite based on the pattern
below. About how much binding does Erin need to
cover the edges of the kite? In kite HJKL,
m?KLP 72, and m?HJP 49.5. Find
each measure. 2. m?LHJ 3. m?PKL
about 191.2 in.
81
18
45
Lesson Quiz Part II
Use the diagram for Items 4 and 5. 4. m?WZY
61. Find m?WXY. 5. XV 4.6, and WY 14.2.
Find VZ. 6. Find LP.
119
9.6
18