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Polygons

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Polygons

Definition:

A closed figure formed by line segments so that each segment intersects exactly two others, but only at their endpoints.

These figures arenot polygons

These figures are polygons

Convex:

No line containing a side of the polygon contains a point in its interior

Concave:

A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

Regular:

A convex polygon in which all interior angles have the same measure and all sides are the same length

Irregular:

Two sides (or two interior angles) are not congruent.

Triangle

3 sides

4 sides

Quadrilateral

5 sides

Pentagon

6 sides

Hexagon

7 sides

Heptagon

8 sides

Octagon

9 sides

Nonagon

10 sides

Decagon

Dodecagon

12 sides

n sides

n-gon

- Regular polygons have:
- All side lengths congruent
- All angles congruent

- Apothem of a polygon: the distance from the center to any side of the polygon.

- We can now subdivide the polygon into triangles.

Equilateral:

A

A

B

C

C

BC

=

3.55

cm

B

BC

=

5.16

cm

G

H

I

HI

=

3.70

cm

Scalene:

A triangle in which all 3 sides are different lengths.

AC = 3.47 cm

AB = 3.47 cm

AB = 3.02 cm

AC = 3.15 cm

Isosceles:

A triangle in which at least 2 sides are equal.

- A triangle in which all 3 sides are equal.

GI = 3.70 cm

GH = 3.70 cm

A triangle in which all 3 angles are less than 90˚.

G

°

76

°

°

57

47

H

I

A

°

44

°

108

°

28

C

B

Acute:

Obtuse:

- A triangle in which one and only one angle is greater than 90˚& less than 180˚

Right:

- A triangle in which one and only one angle is 90˚

Equiangular:

- A triangle in which all 3 angles are the same measure.

polygons

triangles

scalene

isosceles

equilateral

Classification by Sides

with Flow Charts & Venn Diagrams

Polygon

Triangle

Scalene

Isosceles

Equilateral

polygons

triangles

right

acute

equiangular

obtuse

Classification by Angles

with Flow Charts & Venn Diagrams

Polygon

Triangle

Right

Obtuse

Acute

Equiangular

- All quadrilaterals have four sides.
- They also have four angles.
- The sum of the four angles totals 360°
- These properties are what make quadrilaterals alike, but what makes them different?

Two sets of parallel sides

Two sets of congruent sides.

The angles that are opposite each other are congruent (equal measure).

Has all properties of quadrilateral and parallelogram

A rectangle also has four right angles.

A rectangle can be referred to as an equiangular parallelogram because all four of it’s angle are right, meaning they are all 90° (four equal angles).

A rhombus is sometimes referred to as a “slanted square”.

A rhombus has all the properties of a quadrilateral and all the properties of a parallelogram, in addition to other properties.

A rhombus is often referred to as a equilateral parallelogram, because it has four sides that are congruent (each side length has equal measure).

- The square is the most specific member of the family of quadrilaterals. The square has the largest number of properties.
- Squares have all the properties of a quadrilateral, all the properties of a parallelogram, all the properties of a rectangle, and all the properties of a rhombus.
- A square can be called a rectangle, rhombus, or a parallelogram because it has all of the properties specific to those figures.

Unlike a parallelogram, rectangle, rhombus, and square who all have two sets of parallel sides, a trapezoid only has one set of parallel sides. These parallel sides are opposite one another. The other set of sides are non parallel.

One can never assume a trapezoid is isosceles unless they are given that the trapezoid has specific properties of an isosceles trapezoid.

Isosceles is defined as having two equal sides. Therefore, an isosceles trapezoid has two equal sides. These equal sides are called the legs of the trapezoid, which are the non-parallel sides of the trapezoid.

Both pair of base angles in an isosceles trapezoid are also congruent.

- A right trapezoid also has one set of parallel sides, and one set of non-parallel sides.
- A right trapezoid has exactly two right angles. This means that two angles measure 90°.
- There should be no problem identifying this quadrilateral correctly, because it’s just like it’s name. When you think of right trapezoid, think of right angles!

It’s important to have a good understanding of how each of the quadrilaterals relate to one another.

Any quadrilateral that has two sets of parallel sides can be considered a parallelogram.

A rectangle and rhombus are both types of parallelograms, and a square can be considered a rectangle, rhombus, and a parallelogram.

Any quadrilateral that has one set of parallel sides is a trapezoid. Isosceles and Right are two types of trapezoids.

Quadrilateral

Parallelogram

Trapezoid

Rectangle

Rhombus

Isosceles

Trapezoid

Right

Trapezoid

Square