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.
2 sets of opposite congruent angles.
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