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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.