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POLYGONS AND
QUADRILATERALS
BY: MARIANA BELTRANENA 9-5
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 polygon
 Vertex- 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. (n2)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= 1,800interior measure
1,800/12= each interior measure
Find each interior measure.
(4-2)180=360
c + 3c + c + 3c = 360
8c=360
C= 45
Plug in c
m<P = m<R = 45 degrees
m<Q= m<S = 135 degrees.
Theorems of Parallelograms and its converse

if a quadrilateral is a polygon then the opposite
sides are congruent.
examples
•
EXAMPL
ES
Converse: if both pairs of opposite sides are congruent
then the quadrilateral is a polygon
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 congruent
 opposite angles are congruent
 consecutive angles are supplementary
 diagonals bisect each other
 has to be a quadrilateral and sides parallel
 one 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 angles
 Diagonals are perpendicular
 One 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.