Quadrilaterals

Report
6.6 Trapezoids and Kites
Check.4.10 Identify and apply properties and relationships
of special figures (e.g., isosceles and equilateral triangles,
family of quadrilaterals, polygons, and solids).
Spi.3.2 Use coordinate geometry to prove characteristics of
polygonal figures.
Check.4.14 Identify and use medians, midsegments,
altitudes, angle bisectors, and perpendicular bisectors of
triangles to solve problems (e.g., find segment lengths,
angle measures, points of concurrency).
"Even if you're on the right track, you'll get run over if you just sit there." Will Rogers
Trapezoids
A trapezoid is a
quadrilateral with only 1
pair of parallel sides.
The median of a trapezoid
is parallel to the bases
and its measure its ½
the sum of the measure
of the bases
EF = ½(AB + DC)
B
base
A
median
E
F
C
base
D
The diagonals of an isosceles trapezoid are congruent.
Both Pairs of Base Angles are congruent.
A  B and D  C
G
If GI  HJ then
GJ  HI
H
PREPARE FOR CONSTRUCTIONS
I
J
Constructions
• Draw a segment AD.
• Place the compass point at A,
open to the width of AD and draw
an arc above AD.
• Label any point on the arc as B.
• Using the same setting, place the
compass at B and drawn an arc to
the right of B.
• Place the compass at D, draw an
arc to intersect the arc drawn from
B. Label as C.
• Use the straight edge to complete.
B
A
C
D
Constructions - Kite
• Draw a segment RT.
• Choose a compass setting greater
than ½ RT. Place compass at R
and make an arc above and
below the line.
• Increase the compass settings
and repeat with same setting at
point T.
• Find the points where the arcs
intersect and label as Q and S.
• Draw QRST
• What special properties do you
notice about this quadrilateral?
Q
R
T
S
• Symmetric, Sides and Angles, bisectors perpendicular
Use Properties of Kites
A. If WXYZ is a kite, find mXYZ.
Since a kite only has one pair
of congruent angles, which are
between the two noncongruent sides,
WXY  WZY. So, WZY =
121.
mW + mX + mY + mZ = 360 Polygon
Interior Angles
Sum Theorem
73 + 121 + mY + 121 = 360 Substitution
mY = 45
Answer: mXYZ = 45
Simplify.
B. If JKLM is a kite, find KL.
A. 5
B. 6
C. 7
D. 8
Constructions – Median of Trapezoid
• Measure WX, ZY and
MN
• Draw a Trapezoid WXYZ.
• Construct the perpendicular
bisectors of XY and WZ. Label
the midpoints as M and N.
• Draw MN
W
M
Z
• What do you find?
X
N
Y
Objective: Understand and apply the properties of trapezoids be able to solve problems
using the medians of trapezoids.
Trapezoids
B
base
A
A trapezoid is a
quadrilateral with only 1
pair of parallel sides.
Both Pairs of Base Angles
are congruent.
median
E
F
A  B and D  C
The median of a trapezoid
is parallel to the bases
and its measure its ½
the sum of the measure
of the bases
C
base
D
The diagonals of an isosceles
trapezoid are congruent.
EF = ½(AB + DC)
G
I
If GI  HJ then
GJ  HI
H
J
Identify a Trapezoid
J(-18, -1), K(-6, 8), L(18, 1), M (-18, -26)
1. Verify that JKLM is a Trapezoid
2. Is JKLM an isosceles trapezoid?
2 Legs are equivalent making it an
isosceles trapezoid
2 Sides are parallel making it a trapezoid
A(5, 1), B(-3,-1), C(-2, 3) and D(2,4)
Determine if ABCD is a trapezoid
Slope AB = ¼, CD = ¼
Slope AD = -1, Slope BC = 4
2 sides are parallel so it is a Trapezoid
Determine if it is isosceles
BC = √17 and AD= √18
No it is not an isosceles trapezoid
Medians of Trapezoid
QRST is an isosceles trapezoid
with median XY.
• Find TS if QR = 22 and XY = 15
• Find m1, m2, m3, m4 if m1=4a -10 and
m3 = 3a + 32.5
m1+ m3 = 180
4a -10 +3a + 32.5 = 180
XY = ½ (QR + TS)
7a – 22.5 = 180
15 = ½ (22 + TS)
7a = 157.5
30 = 22 + TS
a = 22.5
TS = 8
m1=4(22.5) -10= 80
m3 = 100, m2= 100 m4 = 80
DEFG is an isosceles
trapezoid with median MN
1. Find DG if EF = 20
DG = 40
1. Find m1, m2, m3, m4 if
m1 = 3x + 5 and m3 = 6x – 5
m1 = 65, m2 = 65, m115, m4= 115
Summary
• Trapezoid is quadrilateral with 1 pair of sides
parallel.
• Median of a trapezoid is
– Parallel to the bases and equal to ½ sum of the bases
• For an isosceles trapezoid,
– the legs and the diagonals are congruent
– base angles of a trapezoid are congruent.
• Practice Assignment
– Page 440 8 -26 Even

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