Download ch 5 - ariella and nikki - 2012

MATH PROJECT
Ariella Lindenfeld & Nikki Colona
H
G
3
1
4
2
E

EFGH
EF = HG;FG=EH
EFGH
<E = <G
EFGH
<3 = <4
A parallelogram is a quadrilateral with both
pairs of opposite parallel sides.
 Theorem

2.
3.
F

1.
5-1: Opposite sides of the
parallelogram are congruent.
Theorem 5-2: Opposite angles of a
parallelogram are congruent.
Theorem 5-3: Diagonals of a
parallelogram bisect each other.
EXAMPLE:
y 38
X
120
X equals 120 degrees because of OAC which says that
Opposite Angles are congruent
Y equals 22 because it supplementary to 120
So 60 minus 38 is 22




Theorem 5-4: If both pairs of opposite sides of a quadrilateral
are congruent , then the quadrilateral is a parallelogram.
Theorem 5-5: If one pair of opposite sides of a quadrilateral are
both parallel, 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 5-7: If the diagonals of a quadrilateral bisects each
other, then the quadrilateral is a parallelogram.
<
T
1
Q
S
3
2
M
4
<
R
Theorem
L
M
A
C
B
D
AB=CD
>
>
5-
8: If two lines
are parallel,
then all points
on the line are
equidistant
from the other
line.
Theorem 5-9: If
three parallel lines cut
off congruent segments
on one transversal, then
they cut off congruent
segments on every
transversal.
A
B 1
C 4
X
2
3 Y
5
AX II BY II CZ and AB =
BC
XY = YZ
Z
SECTION 3 – THEOREMS INVOLVING
PARALLEL LINES

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

Is parallel to the third side

Is half as long as the third side
EXAMPLE:
2(4x+3) = 13x+1
8x+6=13x+1
5=5x
X=1
5y+7
6y+3
5y+7 = 6y +3
4=y
13x+1
X=1, Y=4
5-4 SPECIAL PARALLELOGRAMS

Rectangle  is a quadrilateral with four right
angles.


Therefore, every rectangle is a parallelogram.
Rhombus  A quadrilateral with four congruent
sides.

Therefore, every rhombus is a parallelogram.
5-4 SPECIAL PARALLELOGRAMS

Square  a quadrilateral with four right angles and
four congruent sides.

Therefore, every square is a rectangle, a rhombus, and
a parallelogram
*** since rectangles, rhombuses, and squares are
parallelograms, they have all the properties of
parallelograms.
5-4 SPECIAL PARALLELOGRAMS
 Theorem

The diagonals of a rectangle are congruent
 Theorem


5-12
5-13
The diagonals of a rhombus are perpendicular
Theorem 5-14

Each diagonal of a rhombus are perpendicular
5-4 SPECIAL PARALLELOGRAMS

Theorem 5-15

The midpoint of the hypotenuse of a right triangle is
equidistant from the three vertices

Theorem 5-16

If an angle of a parallelogram is a right angle, then the
parallelogram is a rectangle

Theorem 5-17

If two consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus
5-5 TRAPEZOIDS

Trapezoid a quadrilateral with exactly one pair of
parallel sides


Bases  the parallel sides

Legs  the other sides
Isosceles Trapezoid  a trapezoid with congruent
legs

Base angles are congruent
5-5 TRAPEZOIDS

Theorem 5-18

Base angles of an isosceles trapezoid are congruent
*** Median (of a trapezoid )the segment that joins the
midpoints of the legs

Theorem 5-19

The Median of a Trapezoid

Is parallel to the bases

Has a length equal to the average of the base lengths
PRACTICE QUESTIONS:
1.
4x-y
9
2.
X
75
z
y
PRACTICE QUESTION
42
3.
26
3x-2y
4x+y
PROOF:
Statement
Reason
1. Rectangle QRST
RKST
JQST
1.Given
KS =RT
JT = QS
2.
OSC
3. RT =QS
3.
DB
4. JT = KS
4. substitution
T
J
Q
S
R
K