Daniela Morales Leonhardt 9-5

JOURNAL CHAPTER 6.

Describe what a polygon is. Include a discussion about the

parts of a polygon. Also compare and contrast a convex

with a concave polygon. Compare and contrast

equilateral and equiangular. Give 3 examples of each.

TOPIC 1: Polygons

A polygon is a closed plane figure that contains only straight sides.

All polygons contain interior angles, exterior angles, sides, and diagonals.

Diagonals are segments that connect one of the vertices of a polygon with another (non adjacent) vertex in that same shape.

Examples of polygons (with diagonals in blue):

What is a polygon?

Concave: A concave polygon is a polygon with one or more of its vertices pointing inward.

Convex: A polygon with all of its vertices pointing outward.

Concave vs. Convex

An equilateral polygon is a polygon with all equal (equi-) sides (lateral).

An equiangular polygon is a polygon with all equal (equi-) angles (angular).

Equilateral Vs. Equiangular

Explain the Interior angles theorem for polygons. Give

at least 3 examples.

Int. Angles Theorem for Polygons

TOPIC 2:

The Interior Angles Theorem states that A convex polygon with “n” sides has

angle measures equal to

Interior Angles TheoremFor Polygons

(n-2) 180/n

Example 1. How much do the interior angles of a

nonagon measure? (9-2)180/9 7 x 180 = 1260 1260/9 140°

What is the measure of the interior angles of a regular hexagon?

Example 2.

Apply the Steps:• (6-2) 180/6

• 4 x 180 = 720/6• 120

120°

120°

120°

120°

120°

120°

Find the measure of the interior angles of a square.

Example 3.

Apply the Steps• (4-2) 180 / 4

• 2 x 180= 360/4=

90°

Describe the 4 theorems of parallelograms and their

converse and explain how they are used. Give at least 3

examples of each.

TOPIC 3: Parallelograms

If a quadrilateral has two pairs of parallel sides, then it’s a parallelogram.

If a quadrilateral is a parallelogram, then it has two pairs of parallel sides.

Theorem 1

A

A ll BD ll C

Polygon ABCD is aParallelogram

(Ex. 1)

C

D

B

Example 2 & 3.E

D

F

D ll FE ll G

Quadrilateral FED is a parallelogram

GQuadrilateral XYZ is a rectangle.

Rectangles are parallelograms.We can infer that sides W and Y are parallel,

as are X and Z.

Z

Y

X

W

If a quadrilateral’s two pairs of opposite angles are congruent, then it’s a parallelogram.

If a quadrilateral is a parallelogram, then its two opposite angles are congruent.

Theorem 2 & Example 1.

Examples 2, 3, 4, & 5.

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

If the consecutive angles of a quadrilateral are consecutive, then it’s a parallelogram.

Theorem 3 & Example 1.

1

4

3

2

m. Angle 1 + m. Angle 1 = 100°m. Angle 3 + m. Angle 4 = 100 °

Examples 2, 3, 4.

24

24

156

156

104

104 76

76

90 90

9090

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

If a quadrilateral’s diagonals bisect each other, then it’s a parallelogram.

Theorem 4 & Example 1.

Examples 2, 3, and 4.

TOPIC 4: Quadrilaterals Parallelograms

Describe how to prove that a

quadrilateral is a parallelogram.

Include an explanation about theorem 6.10. Give at least 3 examples

of each.

If a quadrilateral’s opposite sides are congruent, then it’s is a parallelogram.

… has opposite sides that are parallel, then it’s a parallelogram.

… has congruent opposite angles, then it’s a parallelogram.

… has supplementary consecutive angles then it’s a parallelogram

If a quadrilateral’s diagonals bisect each other, then it’s a parallelogram.

If a Quadrilateral…

Theorem 6.10 states that if a quadrilateral has a set of sides that are both congruent and parallel, then it must be a parallelogram.

Theorem 6.10

Examples 1-5.

Supp.

Supp.

Supp.

Supp.

Supp.

Supp.

Supp.

Supp.

Supp.

Supp.

Examples 6-10.

TOPIC 5: Rectangles, Rhombuses,And Squares.

Compare and contrast a rhombus with a square

with a rectangle. Describe the rhombus, square and rectangle theorems. Give

at least 3 examples of each.

A parallelogram with: 4 right angles. Congruent diagonals.

If a parallelogram has a right angle then it’s a rectangle.

If a parallelogram has congruent diagonals, then it’s a rectangle.

Rectangles

Theorems:

Examples

A parallelogram with 4 congruent sides. Diagonals are perpendicular.

If a quadrilateral has two congruent consecutive sides, then it is a rhombus.

If a quadrilateral’s diagonals are perpendicular, then it’s a parallelogram.

Rhombuses

Theorems

Examples

A parallelogram that has 4 right angles and four congruent sides.

It’s both a rhombus and a rectangle, so all their properties apply.

Squares

Examples

TOPIC 6: Trapezoids

Describe a trapezoid. Explain

the trapezoidal theorems. Give at least 3 examples

of each.

A quadrilateral with only one pair of congruent opposite sides.

The sides that are parallel are called the bases of a trapezoid, the non-congruent ones are called legs.

If both legs are congruent, then the trapezoid is isosceles.

The midsegment of a trapezoid is parallel to both bases and it’s half the sum of the bases.

The base angles in an isosceles trapezoid are congruent. An isosceles trapezoid’s diagonals are congruent.

Trapezoids

Isosceles Trapezoids

Examples

base

base

leg leg

IsoscelesTrapezoid

TOPIC 7: Kites

Describe a kite. Explain the kite theorems. Give

at least 3 examples of

each.

A quadrilateral with two sets of congruent (and consecutive) sides. (Opposite sides NOT congruent)

The included angles between the congruent sides are called vertex angles.

One diagonal bisects the other. If a quadrilateral is a kite, then its

diagonals are perpendicular. If a quadrilateral is a kite, then 1 pair of

opposite angles is congruent.

Kites

Examples

Vertex Angle

Vertex Angle

Diagonals

Examples

_____(0-10 pts.) Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each.

_____(0-10 pts.) Explain the Interior angles theorem for polygons. Give at least 3 examples.

_____(0-10 pts.) Describe the 4 theorems of parallelograms and their converse and explain how they are used. Give at least 3 examples of each.

_____(0-10 pts.) Describe how to prove that a quadrilateral is a parallelogram. Include an explanation about theorem 6.10. Give at least 3 examples of each.

_____(0-10 pts.) Compare and contrast a rhombus with a square with a rectangle. Describe the rhombus, square and rectangle theorems. Give at least 3 examples of each.

_____(0-10 pts.) Describe a trapezoid. Explain the trapezoidal theorems. Give at least 3 examples of each.

_____(0-10 pts.) Describe a kite. Explain the kite theorems. Give at least 3 examples of each.

_____(0-5 pts.) Neatness and originality bonus _____Total points earned (90 possible)