Download III – 180

Chapter 8
Quadrilaterals
8.1 Angles of Polygons
Angle Measures of Polygons
 We’re going to use Inductive Reasoning to find
the sum of all the interior and exterior angles of
convex polygons.
 We will do this by drawing as many diagonals as
possible from one vertex.
 A diagonal is a segment drawn from non
consecutive verticies.
 We will also use the Angle Sum Theorem that
says…
 The sum of all the interior angles of a triangle
equals 180°.
Triangles
120°
60°
130° 50°
70°
Sum of interior angles
= 180°
110°
Sum of Exterior angles = 360°
Angles of Polygons
# of
Sides
# of
Diags
3
0
4
5
6
n
# of Δ
Sum of
Int <‘s
Sum of
Ext <‘s
1
180
360
Quadrilaterals
2
I
3
1
II
4
6
5
In Triangle I – the sum
of the 3 angles is 180
degrees.
In Triangle II – the sum
of the 3 angles is 180
degrees.
In the Quad – the sum of the 4 angles is 360°
So, what if 3 angles measure 100° and the 4th
measure 60°?
Then the 3 ext. angles measure 80° and the 4th
measures 120°? Sum of four ext angles = 360°
Angles of Polygons
# of
Sides
# of
Diags
3
4
5
6
n
# of Δ
Sum of
Int <‘s
Sum of
Ext <‘s
0
1
180
360
1
2
360
360
Pentagons
Sum of each triangle:
I – 180°
2
1
II – 180°
I
6
III – 180°
Total 540°
II
3
5
4
What if meas of 4 angles
th
7
8
is
100°
each
and
the
5
III
angle is 140°, what is the
9
measure of all ext angles?
Then the meas of 4 ext <‘s is 80° each and the 5th ext
angle is 40°, then the measure of all ext <‘s is 360°
Angles of Polygons
# of
Sides
# of
Diags
# of Δ
Sum of
Int <‘s
Sum of
Ext <‘s
3
0
1
180
360
4
1
2
360
360
5
2
3
540
360
6
n
Angles of Polygons
# of
Sides
# of
Diags
# of Δ
Sum of
Int <‘s
Sum of
Ext <‘s
3
0
1
180
360
4
1
2
360
360
5
2
3
540
360
6
3
4
720
360
n
Angles of Polygons
# of
Sides
# of
Diags
# of Δ
Sum of
Int <‘s
Sum of
Ext <‘s
3
0
1
180
360
4
1
2
360
360
5
2
3
540
360
6
3
4
720
360
n
n-3
n-2
(n-2)180
360
Regular?
What if the polygons are regular?
Then each interior angle is congruent.
Formula for sum of all interior angles is:
(n – 2)180
So, if regular, EACH interior angle
measures:
(n – 2)180/n
If sum of all exterior angles is 360, then:
360/n is the measure of each < if regular.
8.2 Parallelograms
Parallelograms
Definition – A Quadrilateral with two pairs of
opposite sides that are parallel.
Let us see what else we can prove knowing
this definition. ABD  CDB by ASA
1
3
4
2
Parallelograms
Definition – A Quadrilateral with two pairs
of opposite sides that are parallel.
Characteristics:
Each Diagonal divides the Parallelogram into
Two Congruent Triangles.
Parallelograms
A
B
1
3
D
4
2
C
ABD  CDB by ASA
AB  CD and AD  CB by CPCTC
B  D and
A  C by CPCTC
Parallelograms
Definition – A Quadrilateral with two pairs
of opposite sides that are parallel.
Characteristics:
Each Diagonal divides the Parallelogram into
Two Congruent Triangles.
Both Pairs of Opposite Sides are Congruent.
Both Pairs of Opposite Angles are Congruent.
Parallelograms
A
B
1
5
D
3
E
6
4
2
AD  CB ,
C
5
6 and
4 3
AED  CEB by AAS
AE  EC and BE  ED by CPCTC
Parallelograms
Definition – A Quadrilateral with two pairs
of opposite sides that are parallel.
Characteristics:
Each Diagonal divides the Parallelogram into
Two Congruent Triangles.
Both Pairs of Opposite Sides are Congruent.
Both Pairs of Opposite Angles are Congruent.
Diagonals Bisect Each Other.
Consecutive Interior Angles are
Supplementary.
Don’t Confuse Them
Do not confuse the Definition with the
Characteristics.
There is a lot of memorization in this
chapter, be ready for it.
8.3 Tests for Parallelograms
Tests for Parallelograms
There are six tests to determine if a
quadrilateral is a parallelogram.
If one test works, then all tests would
work.
With the definition and five characteristics,
you have six things, right?
Well, it is not that simple…
One characteristic is not a test. It is
replaced with a test.
Tests
 Def: A quad with two pairs of parallel sides.
 Test: If a quad has two pairs of parallel sides,
then it is a parallelogram.
 Char: Diagonals bisect each other.
 Test: If a quad has diagonals that bisect each
other, then it is a parallelogram.
 Char: Both pairs of opposite sides are
congruent.
 Test: If a quadrilateral has two pair of opposite
sides congruent, then it is a parallelogram.
Tests (Con’t)
Both pairs of opposite angles are
congruent.
If a quad has both pairs of opposite angles
congruent, then it is a parallelogram.
All pairs of consecutive angles are
supplementary.
If a quad has all pairs of consecutive
angles supp, then it is a parallelogram.
The one that doesn’t work!
 A diagonal divides the parallelogram into two
congruent triangles.
 If a diagonal divides into two congruent
triangles, then it is a parallelogram.
The other one
This is the test that is not a characteristic.
If one pair of sides is both parallel and
congruent.
This is a parallelogram.
This is a not a para b/c
one pair is sides is congruent
but the other pair of sides is ||
Coordinate Geometry
Sometimes you will be given four
coordinates and you will need to
determine what type of quadrilateral it
makes.
The easiest way to do this is to do the
slope six times. (We’ll start with four times
today).
Find the slope of the four sides and
determine if you have two pairs of parallel
sides.
Example
A ( -2, 3) B ( -3, -1)
C ( 3, 0) D ( 4, 4)
4
A
D
2
-5
C
B
-2
5
mAB=
4/1
mDC=
4/1
mCB=
1/6
mAD=
1/6
Since mAB= mCD and mBC = mAD we have a para!
8.4 Rectangles
Polygon Family Tree
Polygons
Triangles
Quad’s
Trapezoids
Para’s
Pentagons
Kites
Rectangle
Def: A parallelogram with four right angles.
Rectangle
Def: A parallelogram with four right
angles.
Characteristic:
Diagonals are Congruent
Characteristics
A
D
E
C
B
AB  CD BC  BC
ABC  DCB by SAS
AC  BD by CPCTC
B C
Nice to Know Stuff (NTKS)
A
D
E
B
C
We just proved that the diagonals are congruent.
Since this Rect is also a Para – then the
diagonals bisect each other, thus AE, DE, CE and
BE are all congruent. What do you know about
the four triangles?
Rectangle
Definition:
A parallelogram with four right angles.
Characteristic:
Diagonals are Congruent.
NTKS:
The diagonals make four Isosceles
Triangles.
Triangles opposite of each other are
congruent.
Coordinate Geometry
Using coordinate geometry to classify if a
quadrilateral is a rectangle or not is easy
too.
First determine if the quadrilateral is a
parallelogram by doing the slope four
times.
If it is a parallelogram, then determine if
consecutive sides are perpendicular.
Are the slopes of consecutive sides
“opposite signed, reciprocals?”
Example
A ( 0, 5) B ( -1, 1)
C ( 3, 0) D ( 4, 4)
6
A
4
mAB=
4/1
mDC=
4/1
mCB=
-1/4
mAD=
-1/4
D
2
B
-5
C
-2
5
Since mAB= mCD and mBC = mAD we have a para!
mAB and mCB are “opp signed recip” we have rect.
8.5 Rhombi and Squares
Definition
Rhombus – A parallelogram with four
congruent sides.
Characteristics
C
D
2
1
E
A
3
4
B
<3 and <4 are Rt Angles:
By def:
B/C it’s a Para:
DCE  BCE SSS
1
2 CPCTC
3
4 CPCTC
AC | DB :
Rhombus
Def:
A parallelogram with four congruent sides
Characteristics:
Diagonals are angle bisectors of the vertex
angles.
Diagonals are perpendicular.
NTKS:
Diagonals make four right triangles.
All Right triangles are congruent.
Polygon Family Tree
Polygons
Triangles
Quad’s
Trapezoids
Para’s
Rectangles
Pentagons
Kites
Rhombus
Square
Square
A square has two definitions:
A Rectangle with four congruent sides.
A Rhombus with four right angles.
A square has everything that every
polygon in it’s family tree has.
It has all the parts of the definitions,
characteristics and NTKS from Quad’s,
Para’s, Rect’s and Rhombi.
Example
A (-1, 2) B (2, 1)
C (1, -2) D (-2, -1)
2
A
B
-5
5
D
-2
C
mAC=
mDB=
-2/1
1/2
mAB=
-1/3
mDC=
-1/3
m
=
CB
It’s a para, rect,
rhombus so it is m =
AD
a square.
3/1
3/1
Coordinate Geometry
So, if both pairs of opposite sides are
parallel, it is a parallelogram.
If it is a parallelogram with perpendicular
sides, then it is a rectangle.
If it is a parallelogram with perpendicular
diagonals, then it is a rhombus.
If it is a parallelogram with perpendicular
sides and perpendicular diagonals, then it
is a square.
8.6 Trapezoids and Kites
Trapezoids
A trapezoid is a quadrilateral with only one
pair of opposite sides that are parallel.
There are two special trapezoids.
Isosceles Trapezoids
Right Trapezoids.
Trapezoids
Right Traps
Isosc Traps
Names of Parts
1
4
2
3
The parallel
Only one pair
sides are the
of parallel sides
“bases”
The non parallel sides are the “legs”
The angles at the end of each base are
“base angle pairs”
Obviously these angle pairs are supplementary.
Median of Trapezoids
A median of a trapezoid is a segment drawn
from the midpoint of one leg to the midpoint
of the other leg.
The length of the median is m = (b1 + b2)/2
where b1 and b2 are the bases.
Since this is for the Trapezoid, it works for all
the trapezoid’s children.
Right Trapezoid
A right trapezoid is a trapezoid with two
right angles.
Not much else to do with that.
Isosceles Trapezoid
Def:
A trapezoid where the legs are congruent.
Characteristics:
Diagonals are Congruent.
Base angle pairs are congruent.
NTKS:
Opposite triangles made with the legs of the
trap are congruent.
Opposite triangles made with the bases are
similar and isosceles.
Isosceles Trapezoids
Parallel Sides - Bases
Non -Parallel Sides - Legs
Legs - Congruent
Opp Δ’s - Congruent
Opp Δ’s - Similar
Diagonals - Congruent
Kites
Def:
A quadrilateral with two pair of consecutive
sides that are congruent.
Characteristics:
Diagonal that divides the kite into two
congruent triangles is an angle bisector and a
segment bisector.
Diagonal that divides the kites into two
isosceles triangles is not any kind of bisector.
Diagonals are perpendicular.
Kites
1 2
This diagonal is the angle and
segment bisector.
This diagonal is not the angle
and segment bisector.
<1 and <2 are congruent.
<3 and <4 are congruent.
Congruent segments.
4 3
Perpendicular Diagonals