Journal 6 By: Mariana Botran

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 side vertex diagonal diagonal diagonal side side 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. 12 – 2 = 10 10 x 180 = 1800 Sum of the interior angles = 1800° 5 – 2 = 3 3 x 180 = 540 Sum of the interior angles = 540 ° examples: 3 – 2 = 1 1 x 180 = 180 Sum of the interior angles = 180° 1800° 540° 180°

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. 70 110 80 100 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: Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect eachother. One set of cogruent and parallel sides. 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: Diagonals are perpendicular One of the diagonals bisect the other 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.