Transcript Chapter 5 - Angelfire

Mid-Term Exam Review Project
Chapter 5
Quadrilaterals
Chapter 5: Break Down
• 5-1: Properties of Parallelograms
• 5-2: Ways to Prove that Quadrilaterals are
Parallelograms
• 5-3: Theorems Involving Parallelograms
• 5-4 : Special Quadrilaterals
• 5-5: Trapezoids
Chapter 5: Section 1
Properties of Parallelograms
Parallelograms
A parallelogram is a quadrilateral
with both pairs of opposite sides
parallel.
But there are more
characteristics of
parallelograms than just that…
These additional characteristics
are applied in several theorems
throughout the section.
Theorem 5-1
“Opposite sides of a parallelogram
are congruent.”
This theorem states that the sides in a
parallelogram that are opposite of each
other are congruent. This means that in
ABCD, side AB is congruent to side CD
and side AC to side BD.
A
C
B
D
Theorem 5-2
“Opposite Angles of a parallelogram are
congruent”
This theorem states that in a
parallelogram, the opposite angles of a
parallelogram are congruent.
Theorem 5-3
“Diagonals of a parallelogram bisect each
other”
This theorem states that in any
parallelogram the diagonals bisect
each other.
Section 5-2
Ways to Prove that Quadrilaterals are
Parallelograms
5 Ways to Prove that a Quadrilateral
is a Parallelogram
• Show that both pairs of opposite sides are
parallel
• Show that both pairs of opposite sides are
congruent
• Show that one pair of opposite sides are
both congruent and parallel
• Show that both pairs of opposite angles are
congruent
• Show that the diagonals bisect each other
Theorem 5-4
“If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.”
Theorem 5-4
T
Q
S
R
This theorem states that if side TS is congruent to side QR
and side QT is congruent to side SR, then the quadrilateral
is a parallelogram.
Theorem 5-5
“If one pair of opposite sides of a
quadrilateral are both congruent and
parallel, then the quadrilateral is a
parallelogram.”
Theorem 5-5
V
A
L
D
• Theorem 5-5 states that if side VL and side AD are
both parallel and congruent, then the quadrilateral
is a parallelogram.
Theorem 5-6
“If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.”
Theorem 5-6
M
I
K
E
• This theorem states that if angle M is congruent to
angle E, and angle I is congruent to angle K, then
quadrilateral MIKE is a parallelogram.
Theorem 5-7
If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
Theorem 5-7
H
D
A
M
T
• This theorem states that if segment HA is
congruent to segment AT, and segment MA
is congruent to segment AD, that
quadrilateral HDMT is a parallelogram.
Chapter 5: Section 3
• Theorems Involving Parallel
Lines
Theorem 5-8
• “If two lines are parallel, then all
points on one line are equidistant from
the other line.”
All of the points on a line are
equidistant from points on
the perpendicular bisector of
a parallel line.
Theorem 5-9
• “If three parallel lines cut off congruent
segments on one transversal, then they cut
off congruent segments on every
transversal.”
Theorem 5-10
• “A line that contains the midpoint of one
side of a triangle and is parallel to another
side passes through the midpoint of the third
side.”
Theorem 5-11
• “The segment that joins the midpoints of
two sides of a triangle.”
1) is parallel to the third side
2) is half the length of the third side
Section 5-4
Special Parallelograms
Special Parallelograms
• There are four distinct special parallelograms
that have unique characteristics.
• These special parallelograms include:
• Rectangles
• Rhombi
• Squares
Rectangles
• A rectangle is a quadrilateral with four right
angles. Every rectangle, therefore, is a
parallelogram.
Why?
• Every rectangle is a parallelogram because all angles
in a rectangle are right, and all right angles are
congruent.
• If the both pairs of opposite angles in a quadrilateral
are congruent, then the quadrilateral is a
parallelogram.
Additionally:
• If a rectangle is a parallelogram, then it retains
all of the characteristics of a parallelogram. If
this is true then name all congruencies in the
following diagram.
A
S
R
D
E
Solution:
S
A
R
D
E
Rhombi
• A rhombus is a quadrilateral with four
congruent sides. Therefore, every rhombus
is a parallelogram.
Why?
• Every rhombus is a parallelogram because all four
sides are congruent, and therefore both pairs of
opposite sides are congruent.
Squares
• A square is a quadrilateral with four right
angles and four congruent sides. Therefore,
every square is a rectangle, a rhombus, and
a parallelogram.
Why?
• A square is a rectangle a rhombus, and a
parallelogram because all four sides are congruent
and all four angles are right angles.
Theorems
• Theorem’s 5-12 through 5-13 are very
simple and self explanatory. They read as
follows:
• 5-12: The diagonals of a rectangle are
congruent
• 5-13: The diagonals of a rhombus are
perpendicular.
• 5-14: Each diagonal of a rhombus bisects
two angles of the rhombus.
Theorem 5-12
S
A
R
D
E
• According to theorem 5-12, segment SE is
congruent to segment DA
Theorem 5-13
A
B
C
D
• According to theorem 5-13, segment AD is
congruent to segment CB.
Theorem 5-14
A
B
C
D
• According to theorem 5-14, segment AD
bisects angle BAC and angle BCD, and
segment BC bisects angle ABD and angle
ACD.
Theorem 5-15
“The Midpoint of the hypotenuse of a
right triangle is equidistant from the
three vertices.”
Theorem 5-15
A
M
B
According to this theorem, M is equidistant
from points A, B, and C.
C
Theorem 5-16
If an angle of a parallelogram is a
right angle, then the parallelogram is
a rectangle.
Theorem 5-16
M
A
H
T
• In MATH, we know that angle H is a right
angle. This means that all of the angles are right
angles, and that MATH is a rectangle.
(According to Theorem 5-16)
Theorem 5-17
If two consecutive sides of a
parallelogram are congruent, then the
parallelogram is a rhombus.
Theorem 5-17
A
M
T
H
• In MATH, if we know that segment MA is
congruent to segment AT, then we know that
MATH is a rhombus, because segment MA is
congruent to segment TH and segment AT is
congruent to segment MH. Therefore, all sides are
congruent to each other.
Chart of Special Quadrilaterals
Property
Parallelogram
Rectangle
Rhombus
Square
Opp. Sides Parallel
♫
♫
♫
♫
Opp. Sides Congruent
♫
♫
♫
♫
Opp. Angles Congruent
♫
♫
♫
♫
A diag. forms two congruent ∆s
♫
♫
♫
♫
Diags. Bisect each other
♫
♫
♫
♫
Diags are congruent
♫
♫
Diags. are perpendicular
♫
♫
A diag. bisects two angles
♫
♫
All angles are Right Angles
All sides are congruent
♫
♫
♫
♫
Section 5-5
Trapezoids
Trapezoid
• A trapezoid is defined as a quadrilateral
with exactly one pair of parallel sides.
• The parallel sides are called the bases.
• The other sides are the legs
Isosceles Trapezoid
• A trapezoid with congruent legs is called an
Isosceles Trapezoid.
Theorems
• The Following Theorems Concern
Trapezoids and their dimensions.
Theorem 5-18
Base angles of an isosceles trapezoid
are congruent.
H
R
A
I
• This theorem states that trapezoid HAIR is
isosceles, then angle R is congruent to angle
I, and angle H is congruent to angle A.
Theorem 5-19
The Median of a trapezoid:
(1) Is parallel to the bases;
(2) Has a length equal to the average of the
bases.
The Median of a Trapezoid
A
W
N
E
R
• The median of a trapezoid is the segment that
joins the midpoints of the legs.
• In Trapezoid ANDR, EW is the median because it
joins the midpoints, E and W, of the legs.
D
Theorem 5-19
A
W
N
E
R
• According to this theorem, Segment EW (a)
is parallel to AN and DR, and a length equal
to the average of the lengths of AN and DR.
D
Use the Theorems to Complete:
A
W
R
• Given: AN = 10; DR = 20; AR is congruent to ND
• WE =?
• Angle R is congruent to ?
• Angle A is congruent to ?
N
E
D
Solution:
A
W
R
• WE = 15
• Angle R is congruent to Angle D
• Angle A is congruent to Angle N
N
E
D
The End
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Chris O’Connell
Brent Sneider
Jason Fernandez
Mike
Russell Waldman
FIN