Chapter 6: Quadrilaterals

1
Chapter 6 Quadrilaterals

• Fall 2008
• Geometry

2
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.

3
Polygons

• State whether the figure is a polygon.

4
Identifying Polygons
3 sides Triangle
4 sides Quadrilateral
5 sides Pentagon
6 sides Hexagon
7 sides Heptagon
8 sides Octagon
9 sides Nonagon
10 sides Decagon
12 sides Dodecagon
n sides n-gon
5
Polygons

• A polygon is convex if no line that contains a
  side of the polygon contains a point in the
  interior of the polygon

6
Polygons

• A polygon is concave if it is not convex.

7
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.

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Polygons

• Determine if the polygon is regular.

9
Polygons

• A diagonal of a polygon is a segment that joins
  two nonconsecutive vertices.

B
E
L
M
R
10
Polygons

• The sum of the measures of the interior angles of
  a quadrilateral is 360 .
• A B C D 360

A
B
D
C
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Homework 6.1

• Pg. 325 12 34, 37 39, 41 46

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6.2 Properties of Parallelograms

• A parallelogram is a quadrilateral with both
  pairs of opposite sides parallel.

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Theorems about Parallelograms

• If a quadrilateral is a parallelogram, then its
  opposite sides are congruent.
• AB CD and AD BC

A
B
D
C
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Theorems about Parallelograms

• If a quadrilateral is a parallelogram, then its
  opposite angles are congruent.
• A C and D B

A
B
D
C
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Theorems about Parallelograms

• If a quadrilateral is a parallelogram, then its
  consecutive angles are supplementary.
• D C 180 D A 180
• A B 180 B C 180

A
B
D
C
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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
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Examples

• FGHJ is a parallelogram. Find the unknown
  lengths.
• JH _____
• JK _____

5
F
G
F
G
3
K
3
K
J
H
J
H
18
Examples

• PQRS is a parallelogram. Find the angle
  measures.
• m R
• m Q

P
Q
70
S
R
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Examples

• PQRS is a parallelogram. Find the value of x.

P
Q
3x
120
S
R
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6.3 Proving Quadrilaterals are Parallelograms

• For the 4 theorems about parallelograms, their
  converses are also true.

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Theorems

• If both pairs of opposite sides of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

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Theorems

• If both pairs of opposite angles of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

23
Theorems

• If an angle of a quadrilateral is supplementary
  to both of its consecutive angles, then the
  quadrilateral is a parallelogram.

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Theorems

• If the diagonals of a quadrilateral bisect each
  other, then the quadrilateral is a parallelogram.

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One more

• If one pair of opposite sides of a quadrilateral
  are congruent and parallel, then the
  quadrilateral is a parallelogram.

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Examples

• Is there enough given information to determine
  that the quadrilateral is a parallelogram?

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Examples

• Is there enough given information to determine
  that the quadrilateral is a parallelogram?

65
115
65
28
Examples

• Is there enough given information to determine
  that the quadrilateral is a parallelogram?

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How would you prove ABCD ?
A
B
D
C
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How would you prove ABCD ?
A
B
D
C
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How would you prove ABCD ?
A
B
C
D
32
Homework 6.2 6.3

• Pg. 334 20 37
• Pg. 342 9 19, 32 33

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6.4 Rhombuses, Rectangles, and Squares

• A rhombus is a parallelogram with four congruent
  sides.

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Rhombuses, Rectangles, and Squares

• A rectangle is a parallelogram with four right
  angles.

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Rhombuses, Rectangles, and Squares

• A square is a parallelogram with four congruent
  sides and four right angles.

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Special Parallelograms

• Parallelograms

rectangles
rhombuses
squares
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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.

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Theorems

• A parallelogram is a rhombus iff its diagonals
  are perpendicular.

39
Theorems

• A parallelogram is a rhombus iff each diagonal
  bisects a pair of opposite angles.

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Theorems

• A parallelogram is a rectangle iff its diagonals
  are congruent.

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Examples

• Always, Sometimes, or Never
• A rectangle is a parallelogram.
• A parallelogram is a rhombus.
• A rectangle is a rhombus.
• A square is a rectangle.

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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.

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Examples

• MNPQ is a rectangle. What is the value of x?

N
M
2x
Q
P
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6.4 Homework

• Pg. 351 12-21, 25-43

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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.

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Theorems

• If a trapezoid is isosceles, then each pair of
  base angles is congruent.
• A B , C D

A
B
C
D
47
Theorems

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

A
B
C
D
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Theorems

• A trapezoid is isosceles iff its diagonals are
  congruent.
• ABCD is isosceles iff AC BD

A
B
C
D
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Midsegments of Trapezoids

• The midsegment of a trapezoid is the segment that
  connects the midpoints of its legs.

midsegment
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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
D
A
51
Kites

• A kite is a quadrilateral that has two pairs of
  consecutive congruent sides, but opposite sides
  are not congruent.

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Theorems about kites

• If a quadrilateral is a kite, then its diagonals
  are perpendicular.

D
A
C
B
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Theorems about kites

• If a quadrilateral is a kite, then exactly one
  pair of opposite angles are congruent.
• - A C, B D

D
A
C
B
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Name the bases of trap. ABCD
A
D
C
B
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Trapezoid, Isosceles Trap., Kite, or None
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Trapezoid, Isosceles Trap., Kite, or None
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Trapezoid, Isosceles Trap., Kite, or None
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Find the length of the midsegment
7
11
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Find the length of the midsegment
12
6
60
Find the angle measures of JKLM
J
M
44
L
K
61
Find the angle measures of JKLM
J
M
82
L
K
62
Find the angle measures of JKLM
J
M
132
78
L
K
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6.6 Special Quadrilaterals

• quadrilateral
• Kite parallelogram trapezoid
• rhombus rectangle
• square isosceles trapezoid

64
Example 1

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

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Check which shapes always have the given property.
Property Para. Rect. Rhombus Square Kite Trap.
Both pairs of opp. Sides
Exactly 1 pair of opp. Sides
All sides are
Both pairs of opp.
Exactly 1 pair of opp.
All






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Check which shapes always have the given property.
Property Para. Rect. Rhombus Square Kite Trap.
Diagonals are
Diagonals .
Diag. bisect each other

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6.5-6.6 Homework

• Trapezoid worksheet and 6.6 B Worksheet out of
  workbook.

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6.7 Areas of Triangles and Quadrilaterals

• Area of a Rectangle bh
• Area of a Parallelogram bh
• Area of a Triangle ½ bh

h
b
h
b
h
b
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Areas

• Area of a Trapezoid ½ h (b1 b2)
• Area of a Kite ½ d1d2
• Area of a Rhombus ½ d1d2

b1
h
b2
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6.5-6.7 Homework

• Trapezoid worksheet, Practice 6.6 B