POLYGONS AND QUADRILATERALS

BY: MARIANA BELTRANENA 9-5

POLYGONS AND QUADRILATERALS

Polygon

A polygon is a closed figure with more than 3 straight sides which end points of two lines is the vertex.

Parts of Polygons

Sides- each segment that forms a polygonVertex- common end point of two sides.Diagonal- segments that connects any two non

consecutive vertices.

Convex and Concave polygons

Concave- any figure that has one of the vertices caved in.

Convex- any figure that has all of the vertices pointing out.

Equilateral and Equiangular

Equilateral- all sides are congruent.Equiangular- all angles are congruent.

Interior Angles Theorem for Polygons

The sum of the interior angles of a polygon equals the number of the sides minus 2, times 180. (n-2)180. For each interior angle it is the same equation divided by the number of sides.

examples

Find the sum of the interior angle measure of a convex octagon and find each interior angle. (n-2)180 (8-2)180= 1,080. The interior angles

measure 1,080 degrees. 1080/8=135. Each interior angle measures 135 degrees.

Find the sum of the interior angle measure of a convex doda-gon and find each interior angle. (12-2)180= 1,800interior measure 1,800/12= each interior measure

Find each interior measure.(4-2)180=360c + 3c + c + 3c = 3608c=360C= 45Plug in cm<P = m<R = 45 degreesm<Q= m<S = 135 degrees.

Theorems of Parallelograms and its converse

if a quadrilateral is a polygon then the opposite sides are congruent.

examples

• Converse: if both pairs of opposite sides are congruent then the quadrilateral is a polygon

EXAMPLES

Theorems of Parallelograms and its converse

If a quadrilateral has one set of opposite congruent and parallel sides then it is a parallelogram.

This quad. Is a parallelogram because it has one pair of opposite sides congruent and one pair of parallel sides.

This figure is also a parallelogram because it has two pairs of parallel and congruent opposite sides.

• Converse: If one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram

• Examples: Given- KL ll MJ, KL congruent to MJ• Prove- JKLM is a parallelogram• Proof: It is given that KL congruent to MJ. Since KL

ll MJ, <1 congruent to <2 by the alternate interior angles theo. By the reflexive property of congruence, JL congruent JL. So triangle LMJ by SAS. By CPCTC, <3 congruent <4 and JK ll to LM by the converse of the alternate interior angles theo. Since the opposite sides of JKLM are parallel, JKLM is a parallelogram by deff.

Define if the quadrilateral must be a parallelogram.Yes it must be a parallelogram because if one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram.

Theorems of Parallelograms and its converse

If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram

• Converse: If a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.

Theorems of Parallelograms and its converse

If a quadrilateral is a polygon then the opposite angles are congruent.

• Converse: If both pair of opposite angles are congruent then the quadrilateral is a parallelogram.

How to prove a Quadrilateral is a parallelogram.

opposite sides are always congruentopposite angles are congruentconsecutive angles are supplementarydiagonals bisect each otherhas to be a quadrilateral and sides parallelone set of congruent and parallel sides

Square

Is a parallelogram that is both a rectangle and a square. Properties of a square:

4 congruent sides and angles Diagonals are congruent.

rhombus

Is a parallelogram with 4 congruent sides Properties

Diagonals are perpendicular

rectangle

Is any parallelogram with 4 right angles.Theorem: if a a quadrilateral is a rectangle

then it is a parallelogram. Properties

Diagonals are congruent 4 right angles.

Comparing and contrasting

Trapezoids

Is a quadrilateral with one pair of sides parallel and the other pair concave.

There are isosceles trapezoids with the two legs congruent to each other.

Diagonals are congruent.Base angles are congruent.

Trapezoid theorems

Theorems: 6-6-3: if a quadrilateral is an isosceles trapezoid, then

each pair of base angles are congruent. 6-6-4: a trapezoid has a 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.

Trapezoid midsegment theorem: the midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.

Kites

A kite has two pairs of adjacent anglesDiagonals are perpendicularOne of the diagonals bisect each other.

Kite theorems

6-6-1: if a quad. is a kite, then its diagonals are perpen

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