Download Quadrilateral Notes

Unit 7
Quadrilaterals
Polygons
Polygon
• A polygon is formed by three or more
segments called sides
– No two sides with a common endpoint are
collinear.
– Each side intersects exactly two other sides,
one at each endpoint.
– Each endpoint of a side is a vertex of the
polygon.
– Polygons are named by listing the vertices
consecutively.
Identifying polygons
• State whether the figure is a polygon. If
not, explain why.
Polygons are classified by the
number of sides they have
NUMBER
OF SIDES
3
TYPE OF
POLYGON
NUMBER
OF SIDES
TYPE OF
POLYGON
triangle
8
octagon
4
quadrilateral
9
nonagon
5
pentagon
10
decagon
6
hexagon
12
dodecagon
7
heptagon
N-gon
N-gon
Two Types of Polygons:
1. Convex: If a line was
extended from the sides of a
polygon, it will NOT go
through the interior of the
polygon.
Example:
2. Concave:
If a line was
extended from the sides of a
polygon, it WILL go through
the interior of the polygon.
Example:
I. Use the number of sides to tell what kind of polygon the shape is. Then state whether the polygon is
convex or concave.
1.
2.
3.
__________________
__________________
__________________
__________________
__________________
__________________
Regular Polygon
• A polygon is regular if it is equilateral
and equiangular
• A polygon is equilateral if all of its
sides are congruent
• A polygon is equiangular if all of its
interior angles are congruent
Diagonal
• A segment that joins two
nonconsecutive vertices.
II. Use the diagram at the right to answer the following.
4. Name the polygon by the number of sides it has. __________________
5. Polygon ABCDEFG is one name for the polygon. State two other names.
Polygon _______________
Polygon _______________
6. Name all the diagonals that have vertex E as an endpoint.
_____________________________________________
7. Name the nonconsecutive angles to  A . _______________________________
Quadrilateral
• Four sided figure
• The sum of the measures of the interior
angles of a quadrilateral is 360°
2
1
3
4
m1 m2  m3  m4  360
Properties of
Parallelograms
Parallelogram
• A quadrilateral with both pairs of
opposite sides parallel
Theorem 6.2
• Opposite sides of a parallelogram
are congruent.
Theorem 6.3
• Opposite angles of a
parallelogram are congruent
Theorem 6.4
• Consecutive angles of a
parallelogram are supplementary.
m1  m2  180
1
2
3
4
Theorem 6.5
• Diagonals of a parallelogram
bisect each other.
Examples of Problems:
I. Decide whether the figure is a parallelogram. If it is not, explain why not.
1.
2.
3.
________________________
________________________
________________________
________________________
________________________
________________________
II. Use the diagram of the parallelogram to complete the statement.
Refer to parallelogram GHJK to complete each statement.
K
4. JH  ________
5. LH  ________
6. KH  ________
8
J
6
L
G
H
Refer to parallelogram UVWX to complete each statement.
U
7. If XU  15 and UW  28 , find WZ . ___________
8. If mVWX  120 , find mWXU . ___________
9. If mUVW  55 and mVWX  7x  8 , find x. ___________
V
Z
X
W
III. In
ABCD , mC  105 . Find the measure of each angle.
10. mA  ________
11. mD  ________
IV. Find the value of each variable in the parallelogram.
12.
13.
Proving Quadrilaterals are
Parallelograms
Theorem 6.6
To prove a quadrilateral is a
parallelogram:
• Both pairs of opposite sides are
congruent
Theorem 6.7
To prove a quadrilateral is a
parallelogram:
• Both pairs of opposite angles are
congruent.
Theorem 6.8
To prove a quadrilateral is a
parallelogram:
• An angle is supplementary to both of
its consecutive angles.
2
m1  m2  180
3
m2  m3  180
1
4
Theorem 6.9
To prove a quadrilateral is a
parallelogram:
• Diagonals bisect each other.
Theorem 6.10
To prove a quadrilateral
is a parallelogram:
• One pair of opposite sides are
congruent and parallel.
>
>
Are you given enough information to determine whether the quadrilateral is a
parallelogram? Explain.
1.
5.
2.
3.
6.
7.
4.
8.
What value of x will make the polygon a parallelogram?
1.
2.
3.
Types of
parallelograms
Rhombus
• Parallelogram with four congruent
sides.
Properties of a rhombus
• Diagonals of a rhombus are
perpendicular.
Properties of a rhombus
• Each Diagonal of a rhombus bisects a pair
of opposite angles.
Rectangle
• Parallelogram with four right angles.
Properties of a rectangle
• Diagonals of a rectangle are congruent.
Square
• Parallelogram with four congruent
sides and four congruent angles.
• Both a rhombus and rectangle.
Properties of a square
• Diagonals of a square are
perpendicular.
Properties of a square
• Each diagonal of a square bisects a pair of
opposite angles.
45°
45°
45°
45°
45°
45°
45°
45°
Properties of a square
• Diagonals of a square are congruent.
3-Way Tie
Rectangle
Rhombus
Square
Trapezoids
and Kites
Trapezoid
• Quadrilateral with exactly one pair of
parallel sides.
• Parallel sides are the bases.
• Two pairs of base angles.
• Nonparallel sides are the legs.
Base
>
Leg
Leg
>
Base
Isosceles Trapezoid
• Legs of a trapezoid are congruent.
Theorem 6.14
• Base angles of an isosceles trapezoid are
congruent.
>
A
D
>
B
C
A  B , C  D
Theorem 6.15
• If a trapezoid has one pair of congruent
base angles, then it is an isosceles
trapezoid.
>
A
D
>
B
C
ABCD is an isosceles trapezoid
Theorem 6.16
• Diagonals of an isosceles trapezoid are
congruent.
>
A
D
>
ABCD is isosceles if and only if
B
C
AC  BD
Midsegment
of a trapezoid
• Segment that connects the midpoints of its
legs.
Midsegment
Midsegment Theorem
for trapezoids
• Midsegment is parallel to each base and its
length is one half the sum of the lengths of the
bases.
C
B
M
N
A
D
1
MN= 2 (AD+BC)
Kite
• Quadrilateral that has two pairs of
consecutive congruent sides, but opposite
sides are not congruent.
Theorem 6.18
• Diagonals of a kite are perpendicular.
B
A
C
D
AC  BD
Theorem 6.19
• In a kite, exactly one pair of opposite angles are
congruent.
B
A
C
D
A  C , B  D
Pythagorean Theorem
a b  c
2
c
a
b
2
2
Section 6.6
Special Quadrilaterals
Properties of Quadrilaterals
Property
Both pairs of
opposite sides
are congruent
Rectangle Rhombus
X
Diagonals are
congruent
X
X
X
Diagonals are
perpendicular
Square
Kite
X
X
X
X
Diagonals bisect
one another
X
X
X
X
Consecutive
angles are
supplementary
X
X
X
X
X
X
X
X
Both pairs of
opposite angles
are congruent
Trapezoid
X
Properties of Quadrilaterals
• Quadrilateral ABCD has at least one pair
of opposite sides congruent. What kinds of
quadrilaterals meet this condition?
PARALLELOGRAM
RECTANGLE
SQUARE
RHOMBUS
ISOSCELES
TRAPEZOID
Areas of Triangles and
Quadrilaterals
Area Congruence Postulate
• If two polygons are congruent, then they
have the same area.
Area Addition Postulate
• The area of a region is the sum of the
areas of its non-overlapping parts.
Area Formulas
PARALLELOGRAM
A=bh
RECTANGLE
A=lw
SQUARE
As
2
TRIANGLE
A
1
bh
2
Area Formulas
RHOMBUS
d1
d2
1
A  d1 d 2
2
KITE
d1
d2
1
A  d1 d 2
2
Area Formulas
TRAPEZOID
b1
h
b2
1
A  h(b1  b2 )
2