Quadrilaterals

1
Chapter 8

• Quadrilaterals

2
8.1 Angles of Polygons
3
Angle Measures of Polygons

• Were 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.

4
Triangles
120
Sum of interior angles 180
60
130
50
70
110
Sum of Exterior angles
360
5
Angles of Polygons
6
Quadrilaterals
In Triangle I the sum of the 3 angles is 180
degrees.
2
1
6
I
In Triangle II the sum of the 3 angles is 180
degrees.
II
3
5
4
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
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Angles of Polygons
8
Pentagons
Sum of each triangle I 180 II 180 III
180 Total 540
2
1
6
I
II
3
5
4
What if meas of 4 angles is 100 each and the 5th
angle is 140, what is the measure of all ext
angles?
7
8
III
9
Then the meas of 4 ext lts is 80 each and the
5th ext angle is 40, then the measure of all ext
lts is 360
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Angles of Polygons
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Angles of Polygons
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Angles of Polygons
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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 lt if regular.

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8.2 Parallelograms
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Parallelograms

• Definition A Quadrilateral with two pairs of
  opposite sides that are parallel.
• Let us see what else we can prove knowing this
  definition.

1
3
4
2
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Parallelograms

• Definition A Quadrilateral with two pairs of
  opposite sides that are parallel.
• Characteristics
• Each Diagonal divides the Parallelogram into Two
  Congruent Triangles.

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Parallelograms
A
B
1
3
4
D
2
C
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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.

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Parallelograms
A
B
1
3
E
5
6
4
D
2
C
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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.

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Dont Confuse Them

• Do not confuse the Definition with the
  Characteristics.
• There is a lot of memorization in this chapter,
  be ready for it.

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8.3 Tests for Parallelograms
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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.

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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.

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Tests (Cont)

• 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.

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The one that doesnt work!

• A diagonal divides the parallelogram into two
  congruent triangles.
• If a diagonal divides into two congruent
  triangles, then it is a parallelogram.

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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/cone pair is sides is
  congruentbut the other pair of sides is

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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. (Well start with four times today).
• Find the slope of the four sides and determine if
  you have two pairs of parallel sides.

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Example
A ( -2, 3) B ( -3, -1)C ( 3, 0) D ( 4, 4)
mAB
4/1 4/1 1/6 1/6
mDC
mCB
mAD
Since mAB mCD and mBC mAD we have a para!
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8.4 Rectangles
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Polygon Family Tree
Polygons
Quads
Triangles
Pentagons
Paras
Trapezoids
Kites
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Rectangle

• Def A parallelogram with four right angles.

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Rectangle

• Def A parallelogram with four right angles.
• Characteristic
• Diagonals are Congruent

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Characteristics
A
D
E
C
B
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Nice to Know Stuff (NTKS)
A
D
E
C
B

• 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?

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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.

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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?

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Example
A ( 0, 5) B ( -1, 1)C ( 3, 0) D ( 4, 4)
mAB
4/1 4/1 -1/4 -1/4
mDC
mCB
mAD
Since mAB mCD and mBC mAD we have a para!
mAB and mCB are opp signed recip we have rect.
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8.5 Rhombi and Squares
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Definition

• Rhombus A parallelogram with four congruent
  sides.

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Characteristics
By def
C
D
2
B/C its a Para
1
3
E
4
B
A
lt3 and lt4 are Rt Angles
AC DB
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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.

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Polygon Family Tree
Polygons
Quads
Triangles
Pentagons
Paras
Trapezoids
Kites
Rectangles
Rhombus
Square
43
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
  its family tree has.
• It has all the parts of the definitions,
  characteristics and NTKS from Quads, Paras,
  Rects and Rhombi.

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Example
A (-1, 2) B (2, 1)C (1, -2) D (-2, -1)
mAB
-1/3 -1/3 3/1 3/1
mDC
mCB
mAC
-2/1 1/2
Its a para, rect, rhombus so it is a square.
mAD
mDB
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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.

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8.6 Trapezoids and Kites
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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
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Names of Parts
The parallel sides are the bases
2
1
Only one pair of parallel sides
4
3
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.
49
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 trapezoids children.

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Right Trapezoid

• A right trapezoid is a trapezoid with two right
  angles.
• Not much else to do with that.

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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.

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Isosceles Trapezoids
Parallel Sides - Bases
Non -Parallel Sides - Legs
Legs - Congruent
Diagonals - Congruent
Opp ?s - Congruent
Opp ?s - Similar
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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.

54
Kites
This diagonal is the angle andsegment bisector.
2
1
This diagonal is not the angle and segment
bisector.
lt1 and lt2 are congruent.lt3 and lt4 are congruent.
Congruent segments.
4
3
Perpendicular Diagonals