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Journal
Chapter 6
Javier Mendez
Polygons
A 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.
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.
Convex
When all of the vertices are pointing outside, it is a convex polygon.
These figures are convex
polygons.
Equilateral: When all of the sides of a polygon are congruent.
Equiangular: When all of the angles of a shape are congruent.
Interior Angles Theorem
The 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
Parallelograms
A parallelogram is a shape that has two pairs of parallel sides in it.
Properties of Parallelograms:
1)
2)
3)
4)
In a parallelogram, opposite sides are congruent.
In a parallelogram, opposite angles are congruent.
In a parallelogram, consecutive angles are supplementary.
In a parallelogram, diagonals bisect each other.
A
B
C
D
B=D
A=C
D
A
B
E
C
AE = ED
CE = EB
A AND D ARE SUPP.
C AND B ARE SUPP.
C
D
A
B
A
A AND B ARE
SUPP.
C
B
D
How to Prove that Quadrilaterals
are Parallelograms
If 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:
B
A
GIVEN:
DE= 48
EB= 49
E
C
D
IT IS NOT A PARALLELOGRAM
D
B
GIVEN:
DB ll AB
A
D
GIVEN:
AD ll BC
YES IT IS
B
A
IT IS NOT A PARALLELOGRAM BECAUSE
WE DON’ T KNOW IF THEY ARE
CONGRUENT.
C
D
C
C
A
GIVEN: X = 10
ANGLE D = X * 9 + 5
ANGLE B = 85 + X
ANGLE C AND A ARE CONGRUENT TO
EACH OTHER.
B
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 angles
A rectangle is a parallelogram
Its diagonals are congruent
A rhombus is a qudrilateral with 4 congruent sides
It is a parallelogram
Its diagonals are perpendicular
Diagonals bisect a pair of opposite angles
A square is a quadrilateral with all of the sides congruent and all of the angles too
It is a rectangle, a rhombus and a parallelogram.
Tell if the following shapes are squares, rectangles or rhombus
C
D
GIVEN:
ANGLE A AND
D ARE 45
D
GIVEN:
CA = DB
A
C
RECTANGLE
B
IT IS A RHOMBUS
A
B
GIVEN: ONE OF THE
ANGLES = 90
IT IS A SQUARE
GIVEN: A, B, C, D ARE CONGRUENT
C D
A
IT IS A RHOMBUS
B
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 isosceles
When a trapezoid is isosceles, it has congruent diagonals
The 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?
A
Is this an isosceles
trapezoid?
D
Yes
AC > BD
NO
B
C
AD = 3.2 METERS
BC = 2.9 METERS
IS THIS TRAPEZOID
ISOSCELES?
NO
D
B
A
C
WHAT IS THE SUM OF THE TWO BASES?
56
THE SUM OF THE BASES IS 100, WHAT IS THE
LENGHT OF THE MIDESEGMENT?
50
Kites
A kite is quadrilateral with two pairs of consecutive congruent sides.
Its diagonals are perpendicular
One pair of opposite angles is congruent.
25.7
IS THIS A KITE?
IS THIS A KITE?
YES
26
NO
IS THIS A KITE?
GIVEN: X = 8, Y = 4
Y*3
Y+8
YES
IS THIS A KITE?
45
YES
16 + X
X*3