Transcript Algebra I - Denise Kapler

Unit 4
Polygons
Agenda
?
Polygons
What properties of a polygon exist when the
figure is regular versus irregular?
I know
that a regular polygon is equiangular and
equilateral. An irregular polygon is not equiangular nor
equilateral
Regular hexagons
Irregular hexagons
Word Equilateral – means all sides
are of equal length
February 2015
Agenda
?
Polygons
In a convex n-gon, how many degrees are
held by the interior angles?
I know
The interior angles of a triangle sum to 180°.
1
One degree =
of a full rotation.
360
Word Interior angle sum of a convex
polygon=180(n-2)
February 2015
Applet on Geogebra
Agenda
?
Polygons
If a quadrilateral has opposite sides that are ‖
and ≅ then it is a parallelogram. True
I know
If a quadrilateral is a parallelogram, then its
diagonals bisect each other. I can use the
properties of parallelograms to solve for variables.
Word Quadrilateral – 2d figure with 4 sides
February 2015
If a quadrilateral has opposite sides that
are ‖ and ≅ then it is a parallelogram.
True
Solve for x.
Agenda
?
Polygons
When is a quadrilateral is a parallelogram?
that there are 6 conditions and if just one is met
I know then
a quadrilateral is a parallelogram.
1. One pair of opposite sides are parallel and congruent
2. Both pairs of opposite sides are congruent
3. One angle is supplementary to both of its consecutive angles
4. Both pairs of opposite angles are congruent
5. Diagonals bisect each other
Word
Parallelogram – quadrilateral in which
both pairs of opposite sides are parallel.
February 2015
Agenda
?
Polygons
When is a parallelogram a rectangle?
that there are 2 conditions and if just one is met
I know then
a parallelogram is a rectangle.
1. If a parallelogram has congruent diagonals.
2. If a parallelogram ahs one right angle.
Word
Rectangle – parallelogram with 4 right
angles and congruent diagonals.
February 2015
Agenda
?
Polygons
When is a parallelogram a rhombus?
that there are 3 conditions and if just one is met
I know then
a parallelogram is a rhombus.
1. If a parallelogram has one pair of
congruent consecutive sides.
2. If a parallelogram has
perpendicular diagonals
3. If a parallelogram has one
diagonal that bisects a pair of
opposite angles.
Word
Rhombus: a parallelogram with 4
congruent sides, perpendicular diagonals,
diagonals bisect opposite angles
February 2015
Agenda
Polygons
? When is a parallelogram a square?
I know that there are 2 conditions to prove a figure is a
square.
1. If a parallelogram 4 congruent sides and 1 right angle it is a
square.
2. A square has the properties of both a rhombus and a rectangle.
Square: a parallelogram with 4
congruent sides and 4 right angles. Its
diagonals are perpendicular bisectors.
Word The diagonals bisect opposite angles.
February 2015
Agenda
?
Polygons
When is a quadrilateral a kite?
When is a quadrilateral a trapezoid?
I know that a kite is a quadrilateral with 2 pairs of
congruent consecutive sides.
And that a trapezoid has one pair of parallel sides.
Note an isosceles trapezoid has one pair of
congruent sides. Base angles are congruent.
Word
Kite: diamond shape
Trapezoid
Isosceles trapezoid
February 2015
Math Review February 5, 2015
1.
2. Solve for x
and y.
3. Graph and
describe the
bounded figure
𝑦 = 4; 𝑦 = −2
𝑦 = 2𝑥; 𝑦 =
Polygon ~
1. Regular or
irregular
2. Convex or
concave
1
𝑥
2
Parallelogram opposite sides ‖
and ≅. Opposite
∠′ 𝑠 are ≅.
Diagonals bisect
Exterior angle sum of
each other.
polygon = 360
Consecutive angles
degrees
are supplementary.
Math Review February 9, 2015
Given an octagon:
1a. What is the
sum of the exterior
angles?
1b. What is the
measure of one of
the interior
angles?
1c. How many
diagonals can be
drawn from one
vertex?
Diagonal ~ Line
segment drawn from
one vertex to
another
3.
2. Is ABCD a
Graph and
parallelogram? describe the
bounded figure
𝑥=4
𝑥 = −2
𝑦 = 2𝑥
𝑦 = 2𝑥 − 3
∆𝒚
m=
∆𝒙
d=
∆𝒙
𝟐
+ ∆𝒚
𝟐
Parallelogram opposite sides ‖
and ≅. Opposite
∠′ 𝑠 are ≅.
Given an octagon:
1. What is the sum of the
interior angles?
2. How many triangles
are there?
3. Do the triangles meet
in the center of the
octagon or are they
drawn from one
vertex?
4. Solution: multiply the
number of triangles by
180 degrees.
Math Review February 10, 2015
1a. Given an
2. Given
QRST.
octagon, what is
Solve for x and y
the sum of the
exterior angles?
1b. What is the
measure of one
interior angle?
1c. How many
diagonals does an
octagon have?
Parallelogram – opposite
Number of
angles are congruent.
diagonals of
Opposite sides are
congruent
an n-sided
polygon =
Symbol for parallelogram
𝑛(𝑛 − 3)
2
3. Graph and
describe the
bounded figure
𝑦 = 4; 𝑦 = −2
𝑦 = 2𝑥; 𝑦 =
1
𝑥
2
Parallelogram opposite sides
‖ and ≅.
Opposite ∠′ 𝒔
are ≅.
How many
diagonals
does a triangle
have? zero
How many
diagonals
does a square
have? two
How many diagonals
does an octagon
have? twenty
The number of diagonals
of an n-sided polygon is:
𝑛(𝑛 − 3)
2
Math Review February 11, 2015
1a. Given a
2. Given
EDFY.
regular convex
Solve for x and y
nonagon, what is
the measure of
one exterior
angle?
1b. Which regular
polygon has an
exterior angle
measure of 120°?
Consecutive angles in a
Exterior
angles sum to
360°
parallelogram are
supplementary. Opposite
angles are congruent.
Symbol for parallelogram
3.
Graph the
vertices. Is the
bounded figure a
parallelogram?
R: (-4, 3)
S: (4, 5)
T: (3, -4)
U: (-5, -5)
Parallelogram Diagonals bisect
each other; midpoint
of diagonals is same
𝒙𝟏+𝒙𝟐 𝒚𝟏+𝒚𝟐
point. (
,
)
𝟐
𝟐
Math Review February 12, 2015
1a. Given a
2. Given
ABDC.
regular pentagon,
Solve for x, y,
what is the
and z
measure of one
exterior angle?
1b. Which regular
polygon has an
exterior angle
measure of 90°?
Number of
diagonals of
an n-sided
polygon =
𝑛(𝑛 − 3)
2
Consecutive angles in a
parallelogram are
supplementary. Opposite
angles are congruent.
Symbol for parallelogram
3.
Graph the
vertices. Is the
bounded figure a
parallelogram?
F: (-4, 5)
G: (3 ,3)
H: (6, -4)
J: (-1, -2)
Parallelogram Diagonals bisect
each other at
same point.
Math Review February 13, 2015
Given nonagon
1a. What is the
measure of one
interior angle?
1b. What is the
measure of one
exterior angle?
1c. How many
diagonals does
this figure have?
Number of
diagonals of
an n-sided
polygon =
𝑛(𝑛 − 3)
2
2. Given kite WXYZ . 3. Graph the
vertices. What is
Solve for angles
the bounded
W and Y
specific figure?
A: (-2, 4)
B: (2 ,4)
C: (4, 2)
D: (-4, 2)
Trapezoid – one
pair of parallel
sides. Isosceles
Kite has one pair trapezoid – one pair
of opposite sides
of congruent
are congruent.
angles.
Math Review February 17, 2015
Given a 20-gon
1a. What is the
measure of one
interior angle?
1b. What is the
measure of one
exterior angle?
1c. How many
diagonals does
this figure have?
Number of
diagonals of
an n-sided
polygon =
𝑛(𝑛 − 3)
2
2. Given kite ABCD .
Solve for angles B
and D
3. Graph the
vertices. What is
the bounded
specific figure?
A: (6, 7)
B: (16, 31)
C: (34, 31)
D: (50, 7)
Trapezoid – one
pair of parallel
sides. Isosceles
Kite has one pair trapezoid – one pair
of opposite sides
of congruent
are congruent.
angles.
http://www.mathopenref.com/coordtrapezoid.html