Quadrilaterals

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Chapter 6

• Quadrilaterals

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Section 6.1

• Polygons

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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.

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Identifying polygons

• State whether the figure is a polygon. If not,
  explain why.

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Polygons are classified by the number of sides
they have
NUMBER OF SIDES TYPE OF POLYGON
3
4
5
6
7
NUMBER OF SIDES TYPE OF POLYGON
8
9
10
12
N-gon
octagon
triangle
nonagon
quadrilateral
pentagon
decagon
dodecagon
hexagon
heptagon
N-gon
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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
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2. Concave If a line was extended from the sides
of a polygon, it WILL go through the interior of
the polygon.
Example
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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

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Diagonal

• A segment that joins two nonconsecutive vertices.

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Interior Angles of a Quadrilateral Theorem

• The sum of the measures of the interior angles of
  a quadrilateral is 360

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Section 6.2

• Properties of Parallelograms

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Parallelogram

• A quadrilateral with both pairs of opposite sides
  parallel

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Theorem 6.2

• Opposite sides of a parallelogram are congruent.

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Theorem 6.3

• Opposite angles of a parallelogram are congruent

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Theorem 6.4

• Consecutive angles of a parallelogram are
  supplementary.

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Theorem 6.5

• Diagonals of a parallelogram bisect each other.

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Section 6.3

• Proving Quadrilaterals are Parallelograms

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Theorem 6.6To prove a quadrilateral is a
parallelogram

• Both pairs of opposite sides are congruent

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Theorem 6.7 To prove a quadrilateral is a
parallelogram

• Both pairs of opposite angles are congruent.

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Theorem 6.8 To prove a quadrilateral is a
parallelogram

• An angle is supplementary to both of its
  consecutive angles.

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Theorem 6.9 To prove a quadrilateral is a
parallelogram

• Diagonals bisect each other.

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Theorem 6.10 To prove a quadrilateral is a
parallelogram

• One pair of opposite sides are congruent and
  parallel.

gt
gt
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Section 6.4

• Types of
• parallelograms

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Rhombus

• Parallelogram with four congruent sides.

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Properties of a rhombus

• Diagonals of a rhombus are perpendicular.

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Properties of a rhombus

• Each Diagonal of a rhombus bisects a pair of
  opposite angles.

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Rectangle

• Parallelogram with four right angles.

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Properties of a rectangle

• Diagonals of a rectangle are congruent.

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Square

• Parallelogram with four congruent sides and four
  congruent angles.
• Both a rhombus and rectangle.

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Properties of a square

• Diagonals of a square are perpendicular.

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Properties of a square

• Each diagonal of a square bisects a pair of
  opposite angles.

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45
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45
45
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Properties of a square

• Diagonals of a square are congruent.

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3-Way Tie
Rectangle
Rhombus
Square
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Section 6.5

• Trapezoids and Kites

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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
gt
Leg
Leg
gt
Base
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Isosceles Trapezoid

• Legs of a trapezoid are congruent.

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Theorem 6.14

• Base angles of an isosceles trapezoid are
  congruent.

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Theorem 6.15

• If a trapezoid has one pair of congruent base
  angles, then it is an isosceles trapezoid.

gt
A
B
gt
D
C
ABCD is an isosceles trapezoid
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Theorem 6.16

• Diagonals of an isosceles trapezoid are congruent.

gt
ABCD is isosceles if and only if
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Examples on Board

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Midsegment of a trapezoid

• Segment that connects the midpoints of its legs.

Midsegment
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Midsegment Theoremfor trapezoids

• Midsegment is parallel to each base and its
  length is one half the sum of the lengths of the
  bases.

MN (ADBC)
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Examples on Board
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Kite

• Quadrilateral that has two pairs of consecutive
  congruent sides, but opposite sides are not
  congruent.

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Theorem 6.18

• Diagonals of a kite are perpendicular.

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Theorem 6.19

• In a kite, exactly one pair of opposite angles
  are congruent.

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Examples on Board
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Pythagorean Theorem
c
a
b
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Section 6.6

• Special Quadrilaterals

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Properties of Quadrilaterals
Property Rectangle Rhombus Square Trapezoid Kite
Both pairs of opposite sides are congruent
Diagonals are congruent
Diagonals are perpendicular
Diagonals bisect one another
Consecutive angles are supplementary
Both pairs of opposite angles are congruent
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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Properties of Quadrilaterals

• Quadrilateral ABCD has at least one pair of
  opposite sides congruent. What kinds of
  quadrilaterals meet this condition?

PARALLELOGRAM
RECTANGLE
ISOSCELES TRAPEZOID
RHOMBUS
SQUARE
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Section 6.7

• Areas of Triangles and Quadrilaterals

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Area Congruence Postulate

• If two polygons are congruent, then they have the
  same area.

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Area Addition Postulate

• The area of a region is the sum of the areas of
  its non-overlapping parts.

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Area Formulas
TRIANGLE
RECTANGLE
SQUARE
PARALLELOGRAM
Abh
Alw
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Area Formulas
RHOMBUS
KITE
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Area Formulas
TRAPEZOID