JOURNAL CHAPTER 6

BY: Amani Mubarak 9-5

POLYGON A polygon is a closed figure with

connected straight line segments. Depending on how many sides the

polygon has, it will be given its name.

PARTS OF A POLYGON Each polygon includes 3 different sides:

*Side- each segment*vertex- common endpoints of two points*diagonal- segment that connects any two nonconsecutive vertices.

side

vertex

diagonal

vertex vertex

diagonal diagonal

side

side

CONVEX AND CONCAVE POLYGONS In order to determine if a polygon is either concave or convex the

only thing you need to know is that concave polygons are the ones that have atleast one vertex pointing inside. To know if it is a convex, all of the vertexes must be pointing out.

EXAMPLES:

convex

concave

convex

concave

EQUILATERAL AND EQUIANGULAR POLYGONS Every regular polygon is both equilateral

and equiangular. A polygon is equilateral when all sides are congruent, and equiangular when all angles are congruent.

INTERIOR ANGLE THEOREM FOR POLYGONS The sum of the interior angle measures of a

convex polygon with n sides is (n-2)180°

(3-2)180° 1X180=180 180÷3= 34.3

(4-2)180° 2X180= 360 360÷4=90

c°

c°

3c°

3c°

(4-2)180° 2X180=360C +3c+c+3c= 3608c=360C=45

Angle P and R= 45°Angle S and Q= 135°

P

S

Q

R

PARALLELOGRAMS 4 TheoremsTheorem 6-2-1 If a quadrilateral is a parallelogram, then its opposite sides are congruent.converse: If the opposite sides of a quadrilateral

are congruent, then it is a parallelogram.

Theorem 6-2-2 If a quadrilateral is a parallelogram, then its

opposite angles are congruent.Converse:If the opposite angles of a

quadrilateral are congruent, then it is a parallelogram.

Theorem 6-2-3 If a quadrilateral is a parallelogram, then its

consecutive angles are supplementary.Converse:If the consecutive angles of a

quadrilateral are supplementary, then it is a quadrilateral.

Theorem 6-2-4 If a quadrilateral is a parallelogram, then its

diagonals bisect each other.Converse: If the diagonals of a quadrilateral

bisect each other then it is a parallelogram.

How to prove a quadrilateral is a parallelogram In order to prove a quadrilateral is

parallelogram you must know the following properties:

Opposite sides are congruent and parallel.

Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other One set of congruent and parallel sides

Theorems that prove quadrilateral is parallelogram: PROPERTIES OF RECTANGLES 6-4-1: if a quadrilateral is a rectangle, then it is a

parallelogram. 6-4-2: if a parallelogram is a rectangle, then its

diagonals are congruent.

PROPERTIES OF RHOMBUSES 6-4-3: if a quadrilateral is a rhombus,

then ir is a parallelogram. 6-4-4: if a parallelogram is a rhombus,

then its diagonals are perpendicular. 6-4-5: if a parallelogramis a rhombus,

then each diagonal bisects a pair of opposite angles.

Rhombus, Square, Rectangle Rectangle is a parallelogram with 4 right

angles. Diagonals are congruent. Rhombus is a parallelogram with four

congruent sides. Diagonals are perpendicular. Square is a parallelogram that is both a

rectangle and a rhombus. Its four sides and angles are congruent. Diagonals are congruent and perpendicular.

What this three figures have in common is that they are all parallelograms, and have four sides.

More examples…

Rectangle Theorems Theorem 6-5-1

If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

Theorem 6-5-2If the diagonals of a parallelogram are

congruent, then the parallelogram is a rectangle. D E

FG

A

B C

DD

Gco

ngru

entF

E

Rhombus Theorems Theorem 6-5-3

If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Theorem 6-5-4 If the diagonals of a parallelogram are perpendicular, then the paralellogram is a rhombus.

Theorem 6-5-5If one diagonal of a parallelogram bisects a pair of opposite angles, the the parallelogram is a rhombus.

Trapeziod A trapezoid is a quadrilateral with one pair of parallel

sides. Isosceles is a trapezoid with a pair of congruent legs. Diagonals are congruentBase angles (both sets) are congruentOpposite angles are supplementary.

Trapezoid Theorems 6-6-3: if a quadrilateral is an isosceles

trapezoid, then each pair of base angles are congruent.

6-6-4: if a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.

6-6-5: a trapezoid is isosceles if and only if its diagonals are congruent.

Kite A kite is made up of:

Two pairs of congruent adjacent sides. Diagonals are perpendicular One pair of congruent angles. One of the diagonals bisects the other.

Kite Theorems 6-6-1: if a quadrirateral is a kite,

then its diagonals are perpendicular.

6-6-2: if a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

_____(0-10 pts.) Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each. _____(0-10 pts.) Explain the Interior angles theorem for polygons. Give at least 3 examples. _____(0-10 pts.) Describe the 4 theorems of parallelograms and their converse and explain how they are used. Give at least 3 examples of each. _____(0-10 pts.) Describe how to prove that a quadrilateral is a parallelogram.  Give at least 3 examples of each. _____(0-10 pts.) Compare and contrast a rhombus with a square with a rectangle. Describe the rhombus, square and rectangle theorems. Give at least 3 examples of each. _____(0-10 pts.) Describe a trapezoid. Explain the trapezoidal theorems. Give at least 3 examples of each. _____(0-10 pts.) Describe a kite. Explain the kite theorems. Give at least 3 examples of each.