Download polygon - Catawba County Schools

Chapter 6: Quadrilaterals
Fall 2008
Geometry
6.1 Polygons
• A polygon is a closed plane figure that
is formed by three or more segments
called sides, such that no two sides with
a common endpoint are collinear.
• Each endpoint of a side is a vertex of
the polygon.
• Name a polygon by listing the vertices
in clockwise or counterclockwise order.
Polygons
• State whether the figure is a polygon.
Identifying Polygons
3 sides
4 sides
5 sides
6 sides
7 sides
8 sides
9 sides
10 sides
12 sides
n sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Polygons
• A polygon is convex if no line that
contains a side of the polygon contains
a point in the interior of the polygon
Polygons
• A polygon is concave if it is not
convex.
Polygons
• A polygon is equilateral if all of its
sides are congruent.
• A polygon is equiangular if all of its
angles are congruent.
• A polygon is regular if it is equilateral
and equiangular.
Polygons
• Determine if the polygon is regular.
Polygons
• A diagonal of a polygon is a segment
that joins two nonconsecutive vertices.
B
E
L
R
M
Polygons
• The sum of the measures of the interior
angles of a quadrilateral is 360 .
–
A + B + C + D = 360
A
B
C
D
Homework 6.1
• Pg. 325 # 12 – 34, 37 – 39, 41 – 46
6.2 Properties of Parallelograms
• A parallelogram is a quadrilateral with
both pairs of opposite sides parallel.
Theorems about Parallelograms
• If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
– AB = CD and AD = BC
A
D
B
C
Theorems about Parallelograms
• If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
–
A=
C and
D=
B
A
D
B
C
Theorems about Parallelograms
• If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary.
•
•
D + C = 180 ;
A + B = 180 ;
D+
B+
A = 180
C = 180
A
D
B
C
Theorems about Parallelograms
• If a quadrilateral is a parallelogram,
then its diagonals bisect each other.
• AM = MC and DM = MB
A
B
M
D
C
Examples
• FGHJ is a parallelogram. Find the
unknown lengths.
5
– JH = _____
– JK = _____
F
G
K
J
3
H
Examples
• PQRS is a parallelogram. Find the angle
measures.
–m R=
–m Q=
P
Q
70
S
R
Examples
• PQRS is a parallelogram. Find the value
of x.
P
3x
S
Q
120
R
6.3 Proving Quadrilaterals
are Parallelograms
• For the 4 theorems about
parallelograms, their converses are also
true.
Theorems
• If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorems
• If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorems
• If an angle of a quadrilateral is
supplementary to both of its
consecutive angles, then the
quadrilateral is a parallelogram.
Theorems
• If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
One more…
• If one pair of opposite sides of a
quadrilateral are congruent and parallel,
then the quadrilateral is a
parallelogram.
Examples
• Is there enough given information to
determine that the quadrilateral is a
parallelogram?
Examples
• Is there enough given information to
determine that the quadrilateral is a
parallelogram?
65
115
65
Examples
• Is there enough given information to
determine that the quadrilateral is a
parallelogram?
How would you prove ABCD
A
D
?
B
C
How would you prove ABCD
A
D
?
B
C
How would you prove ABCD
A
C
B
D
?
Homework 6.2 – 6.3
• Pg. 334 # 20 – 37
• Pg. 342 # 9 – 19, 32 – 33
6.4 Rhombuses, Rectangles, and Squares
• A rhombus is a parallelogram with four
congruent sides.
Rhombuses, Rectangles, and Squares
• A rectangle is a parallelogram with
four right angles.
Rhombuses, Rectangles, and Squares
• A square is a parallelogram with four
congruent sides and four right angles.
Special Parallelograms
• Parallelograms
rectangles
rhombuses
squares
Corollaries about Special Parallelograms
• Rhombus Corollary
– A quadrilateral is a rhombus iff it has 4
congruent sides.
• Rectangle Corollary
– A quadrilateral is a rectangle iff it has 4
right angles.
• Square Corollary
– A quadrilateral is a square iff it is a
rhombus and a rectangle.
Theorems
• A parallelogram is a rhombus iff its
diagonals are perpendicular.
Theorems
• A parallelogram is a rhombus iff each
diagonal bisects a pair of opposite
angles.
Theorems
• A parallelogram is a rectangle iff its
diagonals are congruent.
Examples
• Always, Sometimes, or Never
–A
–A
–A
–A
rectangle is a parallelogram.
parallelogram is a rhombus.
rectangle is a rhombus.
square is a rectangle.
Examples
• (A) Parallelogram; (B) Rectangle;
(C) Rhombus; (D) Square
– All sides are congruent.
– All angles are congruent.
– Opposite angles are congruent.
– The diagonals are congruent.
Examples
• MNPQ is a rectangle. What is the value
of x?
M
N
2x
Q
P
6.4 Homework
• Pg. 351 # 12-21, 25-43
6.5 Trapezoids and Kites
• A trapezoid is a quadrilateral with
exactly one pair of parallel sides.
– The parallel sides are the bases.
– A trapezoid has two pairs of base angles.
– The nonparallel sides are called the legs.
• If the legs are congruent then it is an isosceles
trapezoid.
Theorems
• If a trapezoid is isosceles, then each
pair of base angles is congruent.
•
A=
B,
A
D
C=
D
B
C
Theorems
• If a trapezoid has a pair of congruent
base angles, then it is an isosceles
trapezoid.
A
D
B
C
Theorems
• A trapezoid is isosceles iff its diagonals
are congruent.
– ABCD is isosceles iff AC = BD
A
D
B
C
Midsegments of Trapezoids
• The midsegment of a trapezoid is the
segment that connects the midpoints of
its legs.
midsegment
Midsegment Theorem
• The midsegment of a trapezoid is
parallel to each base and its length is ½
the sum of the lengths of the bases.
– MN ll AD, MN ll BC, MN = ½ (AD + BC)
B
C
N
M
A
D
Kites
• A kite is a quadrilateral that has two
pairs of consecutive congruent sides,
but opposite sides are not congruent.
Theorems about kites
• If a quadrilateral is a kite, then its
diagonals are perpendicular.
D
A
C
B
Theorems about kites
• If a quadrilateral is a kite, then exactly
one pair of opposite angles are
congruent.
D
A
-
A = C,
B= D
C
B
Name the bases of trap. ABCD
A
D
B
C
Trapezoid, Isosceles Trap., Kite, or None
Trapezoid, Isosceles Trap., Kite, or None
Trapezoid, Isosceles Trap., Kite, or None
Find the length of the midsegment
7
11
Find the length of the midsegment
12
6
Find the angle measures of JKLM
J
M
44
K
L
Find the angle measures of JKLM
J
M
82
K
L
Find the angle measures of JKLM
J
M
132
78
K
L
6.6 Special Quadrilaterals
quadrilateral
Kite
parallelogram
trapezoid
rhombus rectangle
square
isosceles
trapezoid
Example 1
• Quadrilateral ABCD has at least one pair
of opposite sides congruent. What
kinds of quadrilaterals meet this
condition?
Check which shapes always have the
given property.
Property
Both pairs of
~
opp. Sides =
Exactly 1 pair
~
of opp. Sides =
~
All sides are =
Both pairs of
~
opp. =
Exactly 1 pair
~
of opp. =
All
~
=
Para.
Rect.
Rhombus Square
Kite
Trap.
Check which shapes always have the
given property.
Property
Para.
Diagonals are ~
=
Diagonals
Diag. bisect
each other
.
Rect.
Rhombus Square
Kite
Trap.
6.5-6.6 Homework
• Trapezoid worksheet and 6.6 B
Worksheet out of workbook.
6.7 Areas of Triangles and Quadrilaterals
• Area of a Rectangle = bh
h
b
• Area of a Parallelogram = bh
h
b
• Area of a Triangle = ½ bh
h
b
Areas
• Area of a Trapezoid = ½ h (b1 + b2)
b1
h
• Area of a Kite = ½ d1d2
• Area of a Rhombus = ½ d1d2
b2
6.5-6.7 Homework
• Trapezoid worksheet, Practice 6.6 B