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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
created by chris markstrum 2005
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
created by chris markstrum 2005
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
created by chris markstrum 2005
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.
created by chris markstrum 2005
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.
created by chris markstrum 2005
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
created by chris markstrum 2005
, , ,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
Chris Markstrum
click on line DF to have its measureclick on E to have measure angle Fclick on D to get D+E=180 click on D+E= 180 to get measure angle Dclick on C to get measure angle C
created by chris markstrum 2005
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
created by chris markstrum 2005
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
created by chris markstrum 2005
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.