Journal Chapter 6

Javier Mendez

PolygonsA polygon is any shape that is closed, has more than two sides and it does not have curved sides. In a polygon, the segments cannot intersect each other.

Every polygon has sides vertices and they may have diagonals.

diagonal

side

vertex

Not a Polygon

When a polygon has a vertex pointing in, it is a concave polygon.

The flag of Nepal is a concave polygon.

When all of the vertices are pointing outside, it is a convex polygon.

These figures are convex polygons.

Convex

Equilateral: When all of the sides of a polygon are congruent.

Equiangular: When all of the angles of a shape are congruent.

Interior Angles TheoremThe sum of the interior angles of a convex polygon will always be the number of sides the polygon has, minus 2 times 180.

Octagon 8 – 2 = 6 6 x 180 = 1080

1080 degrees

The sum of a 100-gon is: The sum of a 78-gon is:

100 – 2 = 98 x 180 = 17640 degrees 78 – 2 = 76 x 180 = 13680 degrees

ParallelogramsA parallelogram is a shape that has two pairs of parallel sides in it.

Properties of Parallelograms:

1) In a parallelogram, opposite sides are congruent.2) In a parallelogram, opposite angles are congruent.3) In a parallelogram, consecutive angles are supplementary.4) In a parallelogram, diagonals bisect each other.

AB

C D

B = D A = C

AB

CD

E

AE = EDCE = EB

A

B

C

D

A AND D ARE SUPP.

C AND B ARE SUPP.

A

B

C

D

A AND B ARE SUPP.

How to Prove that Quadrilaterals are

ParallelogramsIf a quadrilateral has one pair of congruent and parallel sides, it is a parallelogram.

*The rest of the reasons where shown previpusly.

Tell if the following quadrilaterals are parallelograms:

A

B

C

D

E

GIVEN:DE= 48EB= 49

IT IS NOT A PARALLELOGRAM

A

B

C

D

GIVEN:DB ll AB

YES IT IS

A

B C

D GIVEN:AD ll BC

IT IS NOT A PARALLELOGRAM BECAUSE WE DON’ T KNOW IF THEY ARE CONGRUENT.

A

BC

DGIVEN: X = 10ANGLE D = X * 9 + 5ANGLE B = 85 + XANGLE C AND A ARE CONGRUENT TO EACH OTHER.

IT IS A PARALLELOGRAM

IT IS A PARALLELOGRAM

IT IS NOT A PARALLELOGRAM

Rhombus, Squares and Rectangles

A rectangle is a quadrilateral with 4 right anglesA rectangle is a parallelogramIts diagonals are congruent

A rhombus is a qudrilateral with 4 congruent sidesIt is a parallelogram Its diagonals are perpendicularDiagonals bisect a pair of opposite angles

A square is a quadrilateral with all of the sides congruent and all of the angles tooIt is a rectangle, a rhombus and a parallelogram.

Tell if the following shapes are squares, rectangles or rhombus

AB

CD

GIVEN:CA = DB

RECTANGLEA

B

C

DGIVEN:ANGLE A AND D ARE 45

IT IS A RHOMBUS

GIVEN: ONE OF THE ANGLES = 90

IT IS A SQUARE

GIVEN: A, B, C, D ARE CONGRUENT

IT IS A RHOMBUS

AB

C D

GIVEN: ONE ANGLE IS 90

IT IS A RECTANGLE

Trapezoids

A trapezoid is a quadrilateral that has a pair of parallel sides.In an isosceles trapezoid the pairs of base angles are congruent.When a trapezoid has a pair of congruent base angles, it is isoscelesWhen a trapezoid is isosceles, it has congruent diagonalsThe midsegment of a trapezoid is parallel to both bases and its lenght is one half of the sum od the lenghts of the bases.

Is it isosceles?

Yes

Is this an isosceles trapezoid?

A

C

D AC > BD

NO

B

A

B

C

D

AD = 3.2 METERSBC = 2.9 METERS IS THIS TRAPEZOID ISOSCELES?

NO

28

WHAT IS THE SUM OF THE TWO BASES?

56

THE SUM OF THE BASES IS 100, WHAT IS THE LENGHT OF THE MIDESEGMENT?

50

KitesA kite is quadrilateral with two pairs of consecutive congruent sides.Its diagonals are perpendicularOne pair of opposite angles is congruent.

IS THIS A KITE?

YES

IS THIS A KITE?

NO26

25.7

IS THIS A KITE?GIVEN: X = 8, Y = 4

YES

X * 316 + X

Y * 3 Y + 8

45IS THIS A KITE?

YES