Journal 6

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!!!