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Journal 6 By: Mariana Botran

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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.

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**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

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

Examples:

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

Examples:

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

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

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**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°

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**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.

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**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

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**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. 80 100

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**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.

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**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

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Examples: Yes, parallelogram. We don’t have enough information to tell if it’s a parallelogram. Yes, becuse the diagonals bisect eachother.

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**Rhombus + square + rectangle**

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

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

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

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

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**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

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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

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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)

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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.

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