created by chris markstrum © 2005 state standards for geometry 4: prove basic theorems involving...
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created by chris markstrum © 2005
State Standards for Geometry4: Prove basic theorems involving congruence & similarity.7: Prove & use theorems involving parallel lines and properties of quadrilaterals.12: Find & use side and angle measures of triangles and polygons
6.5 Trapezoids and Kites
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definition
Trapezoid A quadrilateral with exactly one pair of parallel
sides Bases: the parallel sides Legs: the non-parallel sides Isosceles: when the legs are congruent
base1
base2
leg leg
A B
CD
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definition
base1 base2
leg
leg
W
X
YZ
Trapezoid A quadrilateral with exactly one pair of parallel
sides Bases: the parallel sides Legs: the non-parallel sides
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definition
KiteA quadrilateral with two pairs of congruentconsecutive sides.
F
G
H
E
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definition
P
Q
RS
KiteA quadrilateral with two pairs of congruentconsecutive sides.
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A B
theorem
If a trapezoid is isosceles, then each pair of base angles is congruent.
A B
CD
D C
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theorem
If a trapezoid has a pair of congruent base anglesthen it is an isosceles trapezoid.
D C
A B
CD
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theorem
A trapezoid is isosceles if and only ifits diagonals are congruent.
AC BD
A B
CD
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theorem
The midsegment of a trapezoid is parallel toeach base
A B
CD
E F12 ( )EF AB DC
and its length is the average of thelengths of the bases.
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theorem
The diagonals of a kite are perpendicular.
A
B
C
D
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theorem
The angles between the non-congruent sidesof a kite are congruent.
A
B
CD
A C
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, , ,DF m C m D m F Find
ExampleCDEF is an isosceles trapezoid.
C D
EF
CE = 10
95o
10DF CE
95m F m E
180m D m E
85m D
85m C m D
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W
Example
The vertices of WXYZ are W(-1, 2), X(3, 0), Y(4, -3), Z(-4, 1)Show that WXYZ is an isosceles trapezoid.
X
Y
Z
WXm
ZYm
0 2
3 1
1 3
4 4
1
2
1
2
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W
Example
X
Y
Z
1 2
4 1
3 0
4 3
1
3
3
1
2WX ZYm m
The vertices of WXYZ are W(-1, 2), X(3, 0), Y(4, -3), Z(-4, 1)Show that WXYZ is an isosceles trapezoid.
WZm
XYm
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W
Example
X
Y
Z
1
3WZ
m
( ) ( ) 1 4 2 12 2
1 0
( ) ( )3 4 0 32 2
1 0
WZ
XY
3XY
m
The vertices of WXYZ are W(-1, 2), X(3, 0), Y(4, -3), Z(-4, 1)Show that WXYZ is an isosceles trapezoid.
1
2WX ZY
m m
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P Q
RS
Example
PQRS is a trapezoid with the given measurements.
M N
8
10
85o
110o
, ,MN m S m RNM Find
2
PQMN
SR
8 10
2
9
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Example
85o
110o
PQRS is a trapezoid with the given measurements.
// . 'corr s
S PMN m S m PMN
85m S
P Q
RS
M N
8
10
, ,MN m S m RNM Find
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, ,MN m S m RNM Find
Example
PQRS is a trapezoid with the given measurements.
// . 'corr s
Q RNM m Q m RNM
110m RNM 85o
110oP Q
R
M N
8
10
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H
I
J
L
ExampleHIJK is a kite.
Find HP
5P
2 9
5 2 92 2 2 H P ( )
2 5 2 92 H P
H P 2 4
H P 2
2
HPL is a right triangle
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, ,m D m E m F Find
70o
D
E
F
G
ExampleDEFG is a kite.
125o
x + 30
x40o
125o
360m D m E m F m G 125m E m G
30 125 125 360x x
2 280 360x 40x
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Summary What are the characteristics of a
trapezoid?
What are the characteristics of a kite?
Compare a kite and a rhombus.