Unit 6 Quadrilaterals Part 1

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Unit 6QuadrilateralsPart 1

• Parallelograms

Modified by Lisa Palen
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Definition

• A parallelogram is a quadrilateral whose opposite
  sides are parallel.
• Its symbol is a small figure

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Naming a Parallelogram

• A parallelogram is named using all four vertices.
• You can start from any one vertex, but you must
  continue in a clockwise or counterclockwise
  direction.
• For example, this can be either
  ABCD or ADCB.

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Basic Properties

• There are four basic properties of all
  parallelograms.
• These properties have to do with the angles, the
  sides and the diagonals.

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Opposite Sides

• Theorem Opposite sides of a parallelogram are
  congruent.
• That means that .
• So, if AB 7, then _____ 7?

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Opposite Angles

• One pair of opposite angles is ?A and
• ? C. The other pair is ? B and ? D.

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Opposite Angles

• Theorem Opposite angles of a parallelogram are
  congruent.
• Complete If m ? A 75? and m ? B
  105?, then m ? C ______ and m ? D ______ .

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Consecutive Angles

• Each angle is consecutive to two other angles. ?A
  is consecutive with ? B and ? D.

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Consecutive Angles in Parallelograms

• Theorem Consecutive angles in a parallelogram are
  supplementary.
• Therefore, m ? A m ? B 180? and
  m ? A m ? D 180?.
• If mltC 46?, then m ? B _____?

Consecutive INTERIOR Angles are Supplementary!
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Diagonals

• Diagonals are segments that join non-consecutive
  vertices.
• For example, in this diagram, the only two
  diagonals are .

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Diagonal Property

• When the diagonals of a parallelogram intersect,
  they meet at the midpoint of each diagonal.
• So, P is the midpoint of .
• Therefore, they bisect each other so
  and .
• But, the diagonals are not congruent!

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Diagonal Property

• Theorem The diagonals of a parallelogram bisect
  each other.

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Parallelogram Summary

• By its definition, opposite sides are parallel.
• Other properties (theorems)
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.

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Examples

• 1. Draw HKLP.
• 2. Complete HK _______ and HP
  ________ .
• 3. mltK mlt______ .
• 4. mltL mlt______ 180?.
• 5. If mltP 65?, then mltH ____, mltK
  ______ and mltL ______ .

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Examples (contd)

• 6. Draw in the diagonals. They intersect at M.
• 7. Complete If HM 5, then ML ____ .
• 8. If KM 7, then KP ____ .
• 9. If HL 15, then ML ____ .
• 10. If mltHPK 36?, then mltPKL _____ .

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Part 2

• Tests for Parallelograms

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Review Properties of Parallelograms

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.

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How can you tell if a quadrilateral is a
parallelogram?

• Defn A quadrilateral is a parallelogram iff
  opposite sides are parallel.
• Property If a quadrilateral is a parallelogram,
  then opposite sides are parallel.
• Test If opposite sides of a quadrilateral are
  parallel, then it is a parallelogram.

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Proving Quadrilaterals as Parallelograms
Theorem 1
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram .
H
G
Theorem 2
E
F
If one pair of opposite sides of a quadrilateral
are both congruent and parallel, then the
quadrilateral is a parallelogram .
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Theorem
Theorem 3
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
G
H
then Quad. EFGH is a parallelogram.
M
Theorem 4
E
F
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram .
then Quad. EFGH is a parallelogram.
EM GM and HM FM
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5 ways to prove that a quadrilateral is a
parallelogram.
1. Show that both pairs of opposite sides are
. definition
2. Show that both pairs of opposite sides are ? .
3. Show that one pair of opposite sides are both
and ? .
4. Show that both pairs of opposite angles are ? .
5. Show that the diagonals bisect each other .
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Examples
Example 1
Find the values of x and y that ensures the
quadrilateral is a parallelogram.
y2
6x 4x 8 2x 8 x 4
2y y 2 y 2
6x
4x8
2y
Find the value of x and y that ensure the
quadrilateral is a parallelogram.
Example 2
5y 120 180 5y 60 y 12
2x 8 120 2x 112 x 56
5y
(2x 8)
120
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Part 3

• Rectangles

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Rectangles
Definition
A rectangle is a quadrilateral with four right
angles.
Is a rectangle is a parallelogram?
Yes, since opposite angles are congruent.
Thus a rectangle has all the properties of a
parallelogram.

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.

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Properties of Rectangles
If a parallelogram is a rectangle, then its
diagonals are congruent.
Theorem
Therefore, ?AEB, ?BEC, ?CED, and ?AED are
isosceles triangles.
Converse
If the diagonals of a parallelogram are
congruent , then the parallelogram is a rectangle.
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Properties of Rectangles

• Parallelogram Properties
• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
• Plus
• All angles are right angles.
• Diagonals are congruent.
• Also ?AEB, ?BEC, ?CED, and ?AED are isosceles
  triangles

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Examples.

• If AE 3x 2 and BE 29, find the value of x.
• If AC 21, then BE _______.
• If mlt1 4x and mlt4 2x, find the value of x.
• If mlt2 40, find mlt1, mlt3, mlt4, mlt5 and mlt6.

x 9 units
10.5 units
x 18 units
mlt150, mlt340, mlt480, mlt5100, mlt640
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Part 4

• Rhombi
• and
• Squares

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Rhombus
Definition
A rhombus is a quadrilateral with four congruent
sides.

Is a rhombus a parallelogram?

Yes, since opposite sides are congruent.
Since a rhombus is a parallelogram the following
are true

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.

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Rhombus
Note
The four small triangles are congruent, by SSS.
This means the diagonals form four angles that
are congruent, and must measure 90 degrees each.


So the diagonals are perpendicular.
This also means the diagonals bisect each of the
four angles of the rhombus
So the diagonals bisect opposite angles.
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Properties of a Rhombus
Theorem
The diagonals of a rhombus are perpendicular.
Theorem
Each diagonal of a rhombus bisects a pair of
opposite angles.
Note
The small triangles are RIGHT and CONGRUENT!
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Properties of a Rhombus
.
Since a rhombus is a parallelogram the following
are true

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
• Plus
• All four sides are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.
• Also remember the small triangles are RIGHT and
  CONGRUENT!

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Rhombus Examples .....

• Given ABCD is a rhombus. Complete the
  following.
• If AB 9, then AD ______.
• If mlt1 65, the mlt2 _____.
• mlt3 ______.
• If mltADC 80, the mltDAB ______.
• If mlt1 3x -7 and mlt2 2x 3, then x _____.

9 units
65
90
100
10
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Square
Definition
A square is a quadrilateral with four congruent
angles and four congruent sides.
Since every square is a parallelogram as well as
a rhombus and rectangle, it has all the
properties of these quadrilaterals.

• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
• Plus
• Four right angles.
• Four congruent sides.
• Diagonals are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.

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Squares Examples...

• Given ABCD is a square. Complete the following.
• If AB 10, then AD _____ and DC _____.
• If CE 5, then DE _____.
• mltABC _____.
• mltACD _____.
• mltAED _____.

10 units
10 units
5 units
90
45
90
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Part 5

• Trapezoids
• and Kites

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Trapezoid
A quadrilateral with exactly one pair of parallel
sides.
Definition
The parallel sides are called bases and the
non-parallel sides are called legs.
Base
Trapezoid
Leg
Leg
Base
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Median of a Trapezoid
The median of a trapezoid is the segment that
joins the midpoints of the legs. (It is sometimes
called a midsegment.)

• Theorem - The median of a trapezoid is parallel
  to the bases.
• Theorem - The length of the median is one-half
  the sum of the lengths of the bases.

Median
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Isosceles Trapezoid
A trapezoid with congruent legs.
Definition
Isosceles trapezoid
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Properties of Isosceles Trapezoid
1. Both pairs of base angles of an isosceles
trapezoid are congruent.
2. The diagonals of an isosceles trapezoid are
congruent.
B
A
D
C
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Kite
A quadrilateral with two distinct pairs of
congruent adjacent sides.
Definition
Diagonals of a kite are perpendicular.
Theorem
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Flow Chart
Quadrilaterals
Parallelogram
Isosceles Trapezoid
Rhombus
Rectangle
Square