JOURNAL 6..

+

JOURNAL 6..

By: Nina Dorion

+ POLYGONA polygon is a shape with straight sides, a polygon must have at least four angles and cannot have curved sides or an opened one

Polygon, straight sides

Not a polygon, has curved sides

Not a polygon, has one open side.

+Convex and concave polygons:Convex: a polygon that has no angles pointing inwards, no internal angles can be more than 180.

Concave: polygon that has internal angles greater than 180.

+

CONVEX CONCAVE

CONCAVE CONVEX

+EQUILATERAL AND EQUIANGULAR:

equilateral means the sides are congruent

Equiangular means the angles are congruent.

12

12 12

12

Equilateral Equiangular

equilateral

+INTERIOR ANGLES THEOREM FOR POLYGONS

This theorem is used when you want to find the interior angles of a polygon, to do that you use this formula: (n-2)180

For example:When you have a quadrilateral (four sides) you do the following:(4-2)180 fill in for n

4-2=2(2)180=360360/4=90

Each interior angle must be 900

n

+Pentagon 5-2=3180x3=540540/5=108Each angle measures 1080

Hexagon 6-2=4180x4=720720/6=120Each angle measures 1200

8-2=6180x6=10801080/8=135Each angle measures 135o

+

4 theorems of parallelograms

+THEOREM:IF A QUADRILATERAL IS A PARALLELOGRAM, THEN ITS OPPOSITE ANGLES ARE CONGRUENT

CONVERSE:IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM.

+EXAMPLES:

+THEOREM:If a quadrilateral is a parallelogram then its opposite sides are congruent.

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

+EXAMPLES:

+THEOREM:If a quadrilateral is a parallelogram, then its diagonals bisect each other

CONVERSE:If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

+EXAMPLES:

+THEOREM:If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

CONVERSE:If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram

+

45 135

45+135=180

67

113

67+113=180

88

92 88+92=180

+Prove that a quadrilateral is a parallelogram1. Opposite sides are congruent

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supplementary

5. One set of congruent and parallel sides

6. Opposite sides are parallel

+ Opposite sides are congruent:

Opposite angles are congruent:

+Diagonals bisect each other

Consecutive angles are supplementary

a b

M<a+m<b=180

+One set of congruent parallel sides

Opposite sides are parallel

+Rectangle:A parallelogram with 4 right angles

Diagonals are congruent

+Rhombus Parallelogram with 4 congruent sides

+Square

Parallelogram that is both a rectangle and a rhombus

4 congruent sides and congruent angles Diagonals are congruent and perpendicular

+quadrilateral

parallelogram

rectangle rhombus

square

+Trapezoid A quadrilateral with one pair of parallel sides

Isosceles trapezoid: trapezoid with one pair of congruent legs

Properties of isosceles trapezoid:Diagonals are congruentBase angles congruentOpposite angles are supplementary

A

B

M<A+M<B=180

+base

base

legs

+Kite has two pairs of congruent adjacent sides•Diagonals are perpendicular•One pair of congruent angles •One of the diagonals bisect the other

+