Download 4.5, 4.6 and 4.8 - Fulton County Schools

4.5, 4.6 and 4.8
By Tamago and Faris
Today’s Special:
QUADRILATERALS!
• Quadrilateral – a polygon with four sides
o
Parallelogram – a quadrilateral with two pairs of
parallel sides
 Kite – a quadrilateral with exactly two pairs of adjacent
congruent sides
 Rectangle – a quadrilateral with four right angles
 Rhombus – a quadrilateral with four congruent sides
 Square – quadrilateral with four congruent sides and
angles.
o
Trapezoid – a quadrilateral with one and only one
pair of parallel sides
Parallelogram Theorems
• The opposite sides of a parallelogram are
congruent
• The opposite angles of a parallelogram are
congruent
Parallelogram Diagonal Theorems
• A diagonal of a parallelogram divides the
parallelogram into two congruent triangles
• The diagonals of a parallelogram bisect
each other
A
C
ABD  DCA
B
D
OMG, I donno if this is a parallelogram!!
It's a parallelogram...
• If two pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram
• If one pair of opposite sides of a quadrilateral are parallel
and congruent, then the quadrilateral is a parallelogram
• If the diagonals of a quadrilateral bisect each other, then
the quadrilateral is a parallelogram
Rectangle Theorems
• A rectangle is a parallelogram (and so all
parallelogram theorems apply
• The diagonals of a rectangle are congruent
Is It a Rectangle? More Theorems
If one angle of a parallelogram is a right angle, then
the parallelogram is a rectangle
Housebuilder Theorem
If the diagonals of a parallelogram
are congruent, then the
parallelogram is a
rectangle
Theorems of the Rhombus
• A rhombus is a parallelogram (therefore, all
parallelogram theorems apply)
• The diagonals of a rhombus are
perpendicular
Help me! I cannot identify this shape as
a rhombus!
• If one pair of adjacent sides of a parallelogram
are congruent, then the quadrilateral is a
rhombus
• If the diagonals of a parallelogram bisect the
angles of a parallelogram, then the
parallelogram is a rhombus
• If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus
We need squares… NOW!!!!
• A square is a rectangle (so all theorems of
rectangles apply)
• A square is a rhombus (and all theorems of
rhombi apply)
• The diagonals of a square are
congruent and are the
perpendicular bisectors of each
other
Random Kite Slide and Bad Grammar
• The diagonals of a kite are perpendicular
BETWEENNESS POSTULATE
Given points P, Q and R, is PQ + QR = PR, then
Q is between P and R on a line
P
Q
R
Theorems for Triangles!
TRIANGLE MIDSEGMENT THEOREM
A midsegment of a triangle is parallel to a side of a
triangle and its length is equal to half the length of
that side
TRIANGLE INEQUALITY THEOREM
The sum of the length of any two sides of a triangle
is greater than the length of the third side
Interesting Facts that May Catch Your
Attention so You Don’t Fall Asleep… or
Maybe Not.
• The plural of rhombus can either be rhombi
OR rhombuses
• Fun little website about rhombi:
OMFSM CLICK ME!!



Let’s Prove Some Theorems!
A diagonal of a parallelogram divides the parallelogram into two congruent
triangles
Given: parallelogram PLGM with diagonal LM
Prove: LGM  MPL
Parallelogram PLGM with diagonal LM


PL GM
3  2

Given
Def. of parallelogram
Alt. int. angles
PM GL
Def. of parallelogram
1  4
Alt. int. angles
LM LM
Reflexive
LGM  MPL
ASA
Practice Problems - Can You Do
Them?
Are these shapes a parallelogram, rectangle,
rhombus or none?
None
Parallelogram,
Rhombus
Practice Problems - Can You Do
Them?
Find the indicated measures. Assume both are
parallelograms
mSPR
CD
7

5
5

50°
5
60°
40°
7.7
80°