Quadrilaterals

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Quadrilaterals

• Unit 6

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Parallelograms
Rectangles
Quadrilaterals
Squares
Trapezoids

Rhombi
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Quadrilateral
a 4 sided polygon
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Quadrilaterals
parallelogram
rectangle
trapezoid
square
rhombus
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Parallelogram

• A quadrilateral in which opposite sides are
  parallel

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Parallelogram

• has 4 vertices, 4 sides, and
• 2 diagonals (segments connecting non-consecutive
  vertices)

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Characteristics of Parallelograms

• 1. Opposite sides are congruent

Theorem 6.1
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Characteristics of Parallelograms

• 2. Opposite angles are congruent

Theorem 6.2
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Characteristics of Parallelograms

• 3. Consecutive angles are supplementary

M
A
m?M m? H 180 m?H m?T 180
H
T
Theorem 6.3
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Characteristics of Parallelograms

• 4. Diagonals bisect each other

M
A
MU ? UT AU ?UH
U
H
T
Theorem 6.4
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Tests for Parallelograms

• 1. If opposite sides are congruent,

then it is a parallelogram
Theorem 6.5 (converse of 6.1)
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Tests for Parallelograms

• 2. If opposite angles are congruent,

then it is a parallelogram
Theorem 6.6 (converse of 6.2)
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Tests for Parallelograms

• 3. If diagonals bisect each other,

M
A
U
then it is a parallelogram
H
T
Theorem 6.7 (converse of 6.4)
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Tests for Parallelograms

• 4. If one pair of sides is both parallel and
  congruent,

M
A
then it is a parallelogram
H
T
Theorem 6.8
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Tests for Parallelograms

• 5. If two pairs of sides are parallel,

M
A
then it is a parallelogram
H
T
definition
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Rectangle

• A parallelogram with 4 right angles

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Theorem 6-9

• If a parallelogram is a rectangle, then its
  diagonals are congruent

L
O
then LV ? OE
If ?LOVE is a rectangle,
V
E
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Theorem 6-10 (Converse of 6-9)

• If the diagonals of a parallelogram are
  congruent, then it is a rectangle.

L
O
then ?LOVE is a rectangle,
If LV ? OE,
V
E
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Rhombus
a parallelogram with four congruent sides.
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Square
a rectangle with four congruent sides
a rhombus with four right angles
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Properties of Parallelograms
A rectangle is a parallelogram with four right
angles.
A rhombus is a parallelogram with four
congruent sides.
A square is a parallelogram with four congruent
sides and four right angles.
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Quadrilaterals
parallelograms
rhombi
rectangles
squares
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Decide whether the statement is always,
sometimes, or never true.
A rhombus is a rectangle.
Help
SOLUTION
The statement is sometimes true.
In the Venn Diagram, the regions for rhombuses
and rectangles overlap. If the rhombus is a
square, it is a rectangle.
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Decide whether the statement is always,
sometimes, or never true.
A parallelogram is a rectangle.
Help
SOLUTION
The statement is sometimes true.
Some parallelograms are rectangles. In the Venn
diagram, you can see that some of the shapes in
the parallelogram box are in the region for
rectangles, but many arent.
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ABCD is a rectangle. What else do you know about
ABCD ?
SOLUTION
Because ABCD is a rectangle, it has four right
angles by the definition. The definition also
states that rectangles are parallelograms, so
ABCD has all the properties of a parallelogram
Opposite sides are parallel and congruent.
Opposite angles are congruent and consecutive
angles are supplementary.
Diagonals bisect each other.
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ABCD is a rectangle. What else do you know about
ABCD ?
A rectangle is defined as a parallelogram with
four right angles. But any quadrilateral with
four right angles is a rectangle because any
quadrilateral with four right angles is a
parallelogram.
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RHOMBUS COROLLARY
A quadrilateral is a rhombus if and only if it
has four congruent sides.
RECTANGLE COROLLARY
A quadrilateral is a rectangle if and only if it
has four right angles.
SQUARE COROLLARY
A quadrilateral is a square if and only if it is
a rhombus and a rectangle.
You can use these corollaries to prove that a
quadrilateral is a rhombus, rectangle, or square
without proving first that the quadrilateral is a
parallelogram.
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In the diagram, PQRS is a rhombus. What is the
value of y?
SOLUTION
All four sides of a rhombus are congruent, so RS
PS.
5y 6 2y 3
Equate lengths of congruent sides.
5y 2y 9
Add 6 to each side.
3y 9
Subtract 2y from each side.
y 3
Divide each side by 3.
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USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREM 6.10
A parallelogram is a rectangle if and only if
its diagonals are congruent.
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USING DIAGONALS OF SPECIAL PARALLELOGRAMS
Theorem 6.11
If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus.
Conditional statement
Theorem 6.12
If a parallelogram is a rhombus, then its
diagonals are perpendicular.
Converse
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USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREM 6.11 6.12
A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
ABCD is a rhombus if and only if AC BD
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USING DIAGONALS OF SPECIAL PARALLELOGRAMS
THEOREM 6.13
A parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles.
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Trapezoid
a quadrilateral with exactly one pair of parallel
sides.
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Trapezoid
base
leg
leg
base angles
base
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Application of trapezoids turkey calling!
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Isosceles Trapezoid
A trapezoid whose legs are congruent
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Theorem 6-14
Both pairs of base angles of an isosceles
trapezoid are congruent
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Theorem 6-15
The diagonals of an isosceles trapezoid are
congruent
L
P
PA LY
Y
A
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Theorem 6-16
The median of a trapezoid is parallel to the
bases and ½ the sum of the measures of the bases
L
P
E
R
Y
A
RE ½ (PL YA)
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Quadrilaterals
trapezoids