Daniela Morales Leonhardt 9-5 JOURNAL CHAPTER 6.
Jan 19, 2016
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)