Chapter 8

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

Sum of interior angles = 180°60°

50° 70°

Sum of Exterior angles =

110°130°

120°

360°

Angles of Polygons

# of Sides

# of Diags

# of ΔSum of Int <‘s

Sum of Ext <‘s

3 0 1 180 360

4

5

6

n

Quadrilaterals

12

34

6

5II

I

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

# of ΔSum of Int <‘s

Sum of Ext <‘s

3 0 1 180 360

4 1 2 360 360

5

6

n

Pentagons

12

34

6

5II

I

III7 8

9

Sum of each triangle:I – 180°II – 180°III – 180°Total 540°What if meas of 4 angles is 100° each and the 5th angle is 140°, what is the 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 ΔSum 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 ΔSum 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 ΔSum 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.

1

2

3

4

ABD CDB by ASA

Parallelograms

Definition – A Quadrilateral with two pairs of opposite sides that are parallel.

Characteristics:Each Diagonal divides the Parallelogram into

Two Congruent Triangles.

Parallelograms

ABD CDB by ASA

AB CD and AD CB by CPCTC

1

2

3

4

A

C

B

D

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

, 5 6 4 3AD CB and

AED CEB by AAS

1

2

3

4

A

C

B

D

E65

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/cone pair is sides is congruentbut 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

4

2

-2

-5 5

B

A D

C

A ( -2, 3) B ( -3, -1)C ( 3, 0) D ( 4, 4)

mAB=

mDC=

mCB=

mAD=

4/1

4/1

1/6

1/6

Since mAB= mCD and mBC = mAD we have a para!

8.4 Rectangles

Polygon Family Tree

Polygons

Quad’sTriangles Pentagons

Para’sTrapezoids Kites

Rectangle

Def: A parallelogram with four right angles.

Rectangle

Def: A parallelogram with four right angles.Characteristic:

Diagonals are Congruent

Characteristics

AB CD BC BC B C ABC DCB by SAS

C

A D

E

B

AC BD by CPCTC

Nice to Know Stuff (NTKS)

C

A D

E

BWe 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)

mAB=

mDC=

mCB=

mAD=

4/1

4/1

-1/4

-1/4

Since mAB= mCD and mBC = mAD we have a para!

6

4

2

-2

-5 5

B

A

D

C

mAB and mCB are “opp signed recip” we have rect.

8.5 Rhombi and Squares

Definition

Rhombus – A parallelogram with four congruent sides.

Characteristics

DCE BCE SSS

1 2 CPCTC

By def:

B/C it’s a Para:

D C

BA

E

21

43

3 4 CPCTC

<3 and <4 are Rt Angles: 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 TreePolygons

Quad’sTriangles Pentagons

Rectangles Rhombus

Square

Para’sTrapezoids Kites

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

2

-2

-5 5

C

A

D

B

A (-1, 2) B (2, 1)C (1, -2) D (-2, -1)

mAB=

mDC=

mCB=

mAD=

-1/3

-1/3

3/1

3/1

mAC=

mDB=

-2/1

1/2

It’s a para, rect, rhombus so it is a square.

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 TrapezoidsRight Trapezoids.

Right Traps Isosc Traps

Trapezoids

Names of Parts

Only one pair of parallel sides

The parallel sides are the “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.

4 3

21

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

Diagonals - CongruentOpp Δ’s - Congruent

Opp Δ’s - Similar

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

This diagonal is the angle andsegment bisector.

This diagonal is not the angle and segment bisector.

4 3

21

<1 and <2 are congruent.<3 and <4 are congruent.

Congruent segments.

Perpendicular Diagonals