Chapter 8 Quadrilaterals

__Section 8.1
Angles of Polygons__

**Regular polygon**:
has congruent sides (equilateral) and congruent angles (equiangular)

Theorem 8.1: (**Interior
Angle Sum Theorem**): The sum of the measures of the interior
angles of a convex polygon *S= (n - 2)180*

*(n represents the number of sides of the polygon)** *

Theorem 8.2: (**Exterior
Angle Sum Theorem**): The sum of the measures of the exterior
angles (one per vertex) of any convex polygon * = 360*

This can be used to find the
**number of sides**
of a regular polygon if you are given an interior angle.

1. Find the exterior angle (interior and exterior are supplementary, so subtract interior from 180)

2. Divide 360 by the exterior angle measure.

**THE ANSWER MUST BE A WHOLE NUMBER GREATER
THAN 2!!**

**Example:
The measure of an interior angle of a regular polygon is 140.
Find the number of sides.**

**Example:
The measure of an interior angle of a regular polygon is 108.
Find the number of sides.**

n
(# of sides) |
Name of Polygon |
Sum of interior angles |
Sum of exterior angles |

3 | Triangle | ||

4 | Quadrilateral | ||

5 | Pentagon | ||

6 | Hexagon | ||

7 | Heptagon | ||

8 | Octagon | ||

9 | Nonagon | ||

10 | Decagon | ||

11 | 11-gon | ||

12 | Dodecagon |

**Diagonal of a
polygon**: a segment connecting any two nonconsecutive vertices.

__Section 8.2 Parallelograms__

**Parallelogram: **a quadrilateral**
**(4 sides) with 2 sets of
parallel opposite sides.

Th 8.3: opposite sides of a parallelogram are congruent.

Th 8.4: opposite angles of a parallelogram are congruent.

Th 8.5: consecutive angles of a parallelogram are supplementary.

Th 8.6: if a parallelogram has one right angle, then it has 4 right angles.

Th 8.7: The diagonals of a parallelogram bisect each other.

Th 8.8: Each diagonal of a parallelogram separates the parallelogram into two congruent triangles.

EXAMPLES:

__Section 8.3 Tests for
Parallelograms__

** Definition of a Parallelogram**:
If both sets of opposite sides of a quadrilateral
are parallel,
then the quadrilateral is a parallelogram.

** Th 8.9**: If both
pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.

** Th 8.10**: If both
pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.

** Th 8.11**: If the diagonals
of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.

** Th 8.12**: If one
pair of opposite sides of a quadrilateral is both
parallel and congruent, then the quadrilateral is a parallelogram.

**A quadrilateral is a parallelogram if any
one of the following is true.**

**1. Both pairs of opposite sides are
parallel
2. Both pairs of opposite sides are congruent
3. Both pairs of opposite angles are congruent
4. Diagonals bisect each other
5. One pair of opposite sides is both parallel and congruent
**

__Section 8.4 Rectangles__

**Rectangle
(definition)**: a quadrilateral with
4 right angles

**Theorem 8.13**: If a parallelogram is a rectangle,
then the diagonals are congruent.

__Properties of a Rectangle__:

1. Opposite sides are congruent ** and** parallel

2. Opposite angles are congruent

3. Consecutive angles are supplementary

4. Diagonals are congruent ** and** bisect each other

5. All four angles are right angles

__Section 8.5 Rhombi and Squares__

**Rhombus**:
a parallelogram with all four sides congruent

**Th 8.15**: The diagonals of a rhombus
are perpendicular

**Th 8.16**: If the diagonals of a
parallelogram are perpendicular, then the parallelogram is a rhombus

**Th 8.17**: Each diagonal of a rhombus
bisects a pair of opposite angles

**Properties of a Rhombus:**

All the properties of a **
parallelogram**

2 sets of
parallel opposite sides.2 sets of opposite congruent sides.

Consecutive angles are supplementary.

Diagonals bisect each other.

**
AND the properties specific to a rhombus**

All sides are congruent

Diagonals are perpendicular

Diagonals bisect the angles of a rhombus

**Square:**
a quadrilateral that is both a rhombus and a rectangle

**Properties of a Square:**

All the properties of a
**
parallelogram**

2 sets of
parallel opposite sides.

2 sets of opposite congruent sides.

2 sets of opposite congruent angles.

Consecutive angles are supplementary.

Diagonals bisect each other.

All the properties of a **rectangle**

Diagonals are congruent

All four angles are right angles

All the properties of a ** rhombus****
**

All sides are congruent

Diagonals are perpendicular

Diagonals bisect the angles of a rhombus

__Section 8.6 Trapezoids__

**Trapezoid**: a quadrilateral with
__ only
one__ pair of parallel sides

**Parts of a Trapezoid:**

**Bases**: the two parallel sides

**Legs**: the two nonparallel sides

**
Base Angle Pairs**: two angles formed by the same base with different legs

**Isosceles Trapezoid**: a trapezoid with
congruent legs

Th 8.18: both pairs of base angles of an isosceles trapezoid are congruent

Th 8.19: the diagonals of an isosceles trapezoid are congruent

**Median of a
Trapezoid**: a segment
connecting the midpoints of the two legs (also called a midsegment)

Th 8.20: the median (midsegment) of a trapezoid is parallel to the bases and its length is 1/2 the sum of the two bases. (or the average of the two bases)

**Median Formula**:

2 * Median = Base1 + Base2