Download block_24 - Math GR. 9-12

Geometry: From Triangles to
Quadrilaterals and Polygons
MA.912.G.3.2 Compare and contrast
special quadrilaterals on the basis of
their properties.
Block 24
Convex quadrilaterals are classified as
follows:
• Trapezoid (Amer.): exactly one pair of
opposite sides is parallel.
• Parallelogram: both pairs of opposite sides
are parallel.
Convex quadrilaterals are classified as
follows:
• Rhomb (Rhombus): all four sides are of equal
length
• Kite: two adjacent sides are of equal length
and the other two sides also of equal length
Convex quadrilaterals are further
classified as follows:
• Rectangle: all four angles are right angles
• Square: all four sides are of equal length and
all four angles are equal (equiangular), with
each angle a right angle.
Quadrilaterals taxonomy
Properties of quadrilaterals
Parallelograms
Parallelogram: both pairs of opposite sides are
parallel.
Test for Parallelograms
Parallelogram: both pairs of opposite sides are
parallel.
In addition to that definition we have tests to
determine if a quadrilateral is parallelogram
like the following:
Test for Parallelograms
The quadrilateral is a parallelogram:
• If the opposite sides of a quadrilateral are
congruent
Test for Parallelograms
The quadrilateral is a parallelogram:
• If both pairs of opposite angles of a
quadrilateral are congruent
Test for Parallelograms
The quadrilateral is a parallelogram:
• If the diagonals of a quadrilateral bisect each
other
Test for Parallelograms
The quadrilateral is a parallelogram:
• If one pair of opposite sides is parallel and
congruent
Kite
• Kite: two adjacent sides are of equal length
and the other two sides also of equal length.
• This implies that one set of opposite angles is
equal, and that one diagonal perpendicularly
bisects the other.
Rhomb :
• Rhomb: all four sides are of equal length.
• This implies that opposite sides are parallel,
opposite angles are equal, and the diagonals
perpendicularly bisect each other.
Rectangle
• Rectangle (or Oblong): all four angles are
right angles.
• This implies that opposite sides are parallel
and of equal length, and the diagonals bisect
each other and are equal in length.
Square
Square (regular quadrilateral): all four sides are
of equal length (equilateral), and all four
angles are equal (equiangular), with each
angle a right angle.
This implies that opposite sides are parallel (a
square is a parallelogram), and that the
diagonals perpendicularly bisect each other
and are of equal length. A quadrilateral is a
square if and only if it is both a rhombus and a
rectangle.
Trapezoid:
• Trapezoid (Amer.): exactly one pair of
opposite sides is parallel.
• The parallel sides are called bases, and the
nonparallel sides are called legs
• If the legs are congruent then the trapezoid is
called isosceles trapezoid.
Test for Parallelograms and
Coordinates
If the quadrilateral is graphed on the coordinate
plane you can use Distance Formula, Slope
Formula and Midpoint Formula.
The Slope Formula is used to determine if the
opposite sides are parallel the Distance
Formula is used to test opposite sides for
congruency, the Midpoint Formula can be
used to determine if the diagonals are
bisecting each other
Example:
• We will check if a given quadrilateral on the
coordinate plane is a parallelogram
Question:
Prove that quadrilateral ABCD where
• A= (-1,-1)
• B=(3,0)
• C=(4,2)
• D=(0,1)
• Is a parallelogram
Proof:
We will prove that AB||CD
and AD||BC (opposite
sides of quadrilateral
are parallel)
We will use the slope
formula:
y2  y1
m
x2  x1
Proof:
We will prove that AB||CD
and AD||BC
We will use the slope
formula:
0  (1) 1
slopeAB 

3  (1) 4
1  2 1 1
slopeCD 


04 4 4
Proof:
We will prove that AB||CD
and AD||BC
We will use the slope
formula:
1  (1) 2
slopeAD 
 2
0  (1) 1
20 2
slopeBC 
 2
43 1
Proof:
We proved that both pairs
of slopes are the same
so: AB||CD and
AD||BC hence the
quadrilateral ABCD is a
parallelogram
Now answer questions from the
handout
• Find the diagonals of
the parallelogram
• Find the intersection
point of the diagonals
• Verify that the
diagonals bisect each
other
Review and discussion
• Create the Venn diagram and ,,family tree” of
quadrilaterals
• Discus properties of quadrilaterals