By: Mariana Botran

Journal 6

Polygons

• A Polygon is a closed plane figure with three or more straight sides.

examples:

The siluette of that house is a polygon because it is a closed figure with no curved lines.

• The parts of a polygon are:-sides: each segments-vertices: are the common endpoints of two

points-diagonals: segment that connects any two

nonconsecutive vertices.vertex

diagonal

side

vertex

diagonal

side

side

diagonal

vertex

• A convex polygon is a polygon in which all vertices are pointing out.

Examples:

• A concave polygon is a figure that has one or more vertices pointing in.

Examples:

• Equilateral is when all the sides of the polyogon are congruuent.

• Equiangular is when all the angles of a polygon are congruent.

Interior Angle Theorem for Polygons:

• The number of sides minus two and then multiplied times 180 will tell you the sum of the interior angles.

3 – 2 = 11 x 180 = 180Sum of the interior angles = 180°

180°

examples:

5 – 2 = 33 x 180 = 540Sum of the interior angles = 540 °

540°

12 – 2 = 1010 x 180 = 1800Sum of the interior angles = 1800°

1800°

4 theorems of parallelograms and its converse:

• Theorem 6-2-1: If a quadrilateral is a parallelogram,then its opposite sides are congruent

Converse: If a quadrilateral has its opposite sides that are congruent, then it is a parallelogram.

• Theorem 6-2-2: If a quadrilateral is a parallelogram,then its opposite angles are congruent.

Converse: If a quadrilateral has opposite angles that are congruent, then it is a parallelogram

• Theorem 6-2-3: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Converse: If a quadrilateral has consecutive angles that are supplementary, then it is a parallelogram.

100 80 70 110

60 120

• Theorem 6-2-4: If a quadrilateral is a parallelogram, then its diagonals bisect each other.Converse: If a quadrilateral has diagonals that bisect eachother, then it is a parallelogram.

How to prove that a quadrilateral is a parallelogram.

• To know that a quadrilateral is a parallelogram, we have to know the six characteristics of a parallelogram:

1. Opposite sides are congruent.2. Opposite angles are congruent.3. Consecutive angles are supplementary.4. Diagonals bisect eachother.5. One set of cogruent and parallel sides.6. Definition: quadrilateral that has opposite sides

parallel to eachother

Examples:

Yes, parallelogram.

We don’t have enough information to tell if it’s a parallelogram.

Yes, becuse the diagonals bisect eachother.

Rhombus + square + rectangle

• Rhombus: It has 4 congruent sides and diagonals are perpendicular.

• Square: It is equiangular and equilateral, it is both a rectangle and a rhombus and its diagonals are congruent and perpendicular.

• Rectangle: Diagonals are congruent and has four right angles.

These three figures have in common that they are all parallelograms and have four sides.

Trapezoid and its theorems

• A trapezoid is quadrilateral with one pair of parallel sides, each of the parallel sides is called a base and the nonparallel sides are called legs.

Base legs

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

pair of base angles are congruent.

• 6-6-4: if a trapezoid has one 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

Kite• A kite is a quadrilateral that has 2 pairs of congruent adjacent sides (2

lines at the top are congruent and the 2 lines at the bottom are congruent)

• The properties:1. Diagonals are perpendicular2. One of the diagonals bisect the other3. One pair of congruent angles (the ones formed by the non-congruent

sides)

Kite Theorems:• 6-6-1: if a quadrirateral is a kite, then

its diagonals are perpendicular.

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