Journal 6 Jaime Rich
Feb 15, 2016
Journal 6
Jaime Rich
Polygons:
Polygon Parts:
A polygon is any closed shape with straight edges, or sides.
• Side: a segment that forms a polygon• Vertex: common endpoint of sides.• Diagonal: segment that connects 2 non-consecutive vertices.
EX: a
c
b
d
e
abcde is a polygon
a
bc
abc is a pollygon
diagonal
side
vertex
Convex:
Concave:
• All vertices are pointing out• ALL regular polygons are convex
• One ore more vertices are pointing in
EX: Convex polygons
Concave polygons
Equilateral:
Equiangular:
• When all sides in a polygon are congruent
• When all angles in a polygon are congruent
EX: Equilateral Polygons
Equiangular Polygons
Interior Angle Theorem for Polygons:
To know how to find the measure of the angles of a polygon you use this formula:(n-2)180.
n stands for the number of sides each polygon has for example, a rectangle has 4 Sides so the formula is 4-2=2 times 180=360. The sum of all angles in a rectangleIs 360.
To find the measure of each angle, divide the answer you get using the formula Above, by n, or the number of sides. For the rectangle it would be 360/4=90.
EX:(4-2)180=360
All angles
All angles
All angles
Each angle
Each angle
360/4=90
(5-2)180=540
540/5=108
(6-2)180=720
Four Theorems of Parallelograms and
Their Converse
Theorems:
Converse:
If a quadrilateral is a parallelogram then its opposite sides are congruent.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
EX:
Theorems:
Converse:
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadriliateral is a parallelogram.
EX:
Theorems:
Converse:
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
EX:
Theorems:
Converse:
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
EX:50 130
60120
70 110
Proving Quadrilaterals are Parallelograms:
• Opposite angles are congruent• Opposite sides are congruent• Consecutive angles are supplementary• Diagonals bisect each other• Opposite sides are parallel • One set of congruent and parallel sides
EX:
50 130
Consecutive angles are supplementary
Diagonals bisect each other
Congruent sidesCongruent Opposite Angles
Opposite sides are parallel. One set of congruent and parallel sides.
Rectangle:
Theorem:
A parallelogram with four right angles
Diagonals are congruent
EX:
Square:
Theorem:
A parallelogram that is both a rhombus and a rectangle.
All four sides and all four angles are congruent
Diagonals are congruent and perpendicular bisectors of each other
EX:
Rhombus:
Theorem:
A parallelogram with four congruent sides
Diagonals are perpendicular
EX:
Rectangle
Square Rhombus
• Diagonals are congruent
• 4 congruent angles
• Always regular polygon• Sort of mixture between rhombus andrectangle
• Diagonals are perpendicular
• 4 congruent sides
• Polygon• Quadrilateral• Parallelogram• Diagonals bisect each other
Theorems:
Trapezoid:A quadrilateral with a pair of parallel sides
Isosceles trapezoid: one with a pair of congruent legs
• Diagonals are congruent• Base angles are congruent (both sets)• Opposite angles are supplementary
EX:Isosceles Trapezoid
Both labeled angles are supplementaryto each other.
Theorems:
Kite:A quadrilateral with 2 different pairs of congruent sides.
• Two pairs of congruent adjacent sides• Diagonals are perpendicular• One pair of congruent angles (the ones formed by the non-congruent sides)• One of the diagonals bisects the other diagonal
EX:
THE END!!!