Angles of Polygons
Angles
Section 8-1
Interior Angle Sum Thm.
Interior
If a convex polygon has n
sides and S is the sum of
the measures of its interior
angle, then S = 180(n-2).
angle,
Exterior Angle Sum Thm.
Exterior
If a polygon is convex, then
the sum
Geometric
Probability
Section 7-8
Length Probability Postulate
If a point on AB is chosen at
random and C is between A
and B, then the probability
that the point is on AC is
length of AC
length of AB
Area Probability Postulate
If a point in region A is ch
Areas of Circles
and Sectors
and
Section 7-7
Area of a Circle
Area
r
A =r
2
Area of Sector of a Circle
Area
A
O
B
m AB
2
Area of sector AOB =
r
360
Joke Time
Joke
What flower grows between your
nose and your chin?
Tulips
Why were Goldilocks and the Big
B
The Law of Sines and
Law of Cosines
Sections 7-6 and
7-7
Law of Sines
sin A sin B sin C
=
=
a
b
c
c
A
B
a
C
Law of Cosines
a = b + c 2bc cos A
2
2
2
b = a + c 2ac cos B
2
2
2
c = a + b 2ab cos C
2
2
2
c
A
B
a
C
Joke Time
Why are toucans always in debt?
Be
Area of Regular
Polygons
Section 7-5
apothem a segment that is
drawn from the center of a
regular polygon perpendicular
to a side of the polygon
Area of a Regular Polygon
If a regular polygon has an area
of A square units, a perimeter of
P units, and an a
Angles of Elevation
and Depression
Section 7-5
angle of elevation an angle
formed by a horizontal line
and the line of sight to an
object above the level of the
horizontal.
angle of elevation
angle of depression an
angle formed by a horizontal
line and th
Trigonometry
Section 7-4
A trigonometric ratio is a ratio of
sides of a right triangle.
The three most common ratios are
sine, cosine, and tangent.
SOH CAH - TOA
opposite
sine = sin =
hypotenuse
cosine = cos =
adjacent
hypotenuse
opposite
tangent = tan =
Area of Trapezoids,
Rhombi, and Kites
Section 7-4
Area of a Rhombus or Kite
1
A = d1d 2
2
d1
d2
Where d1 and d2 are diagonals.
Area of a Trapezoid
b1
1
A = h(b1 + b2 )
2
h
b2
Joke Time
What is Beethoven doing in
his grave?
De-composing
Why does a lobster
Special Right
Triangles
Triangles
Section 7-3
45 -45 -90 triangle
45
o
o
o
In a 45 -45 -90 triangle, the
hypotenuse is 2 times as long
as a leg.
O
32
45O 2 x
3 45
x
o
x
o
45O
o
3
45O
30 -60 -90 triangle
30
o
o
o
In a 30o-60o-90o triangle, the
hypotenuse i
Special Right
Triangles
Triangles
Section 7-3
45 -45 -90 triangle
45
o
o
o
In a 45 -45 -90 triangle, the
hypotenuse is 2 times as long
as a leg.
O
32
45O 2 x
3 45
x
o
x
o
45O
o
3
45O
30 -60 -90 triangle
30
o
o
o
In a 30o-60o-90o triangle, the
hypotenuse i
The Pythagorean
Thm. & Its Converse
Thm.
Section 7-2
Pythagorean Theorem
Pythagorean
In a right triangle, the sum of the
squares of the measures of the legs
equals the square of the measure of
the hypotenuse.
c
2
2
2
a
a +b =c
b
Converse is also true.
Pyt
Area of
Parallelograms
Section 7-1
Area of a Rectangle
A = bh
Where b = base and
h= height
Area of a Parallelogram
A = bh
Where b = base and
h= height
Area of Triangle
1
A = bh
2
Where b = base and
h= height
Postulate 10-1
The area of a region is the
sum
Geometric Mean &
the Pythagorean Thm.
the
Section 7-1 & 7-2
The geometric mean between 2
positive numbers a and b is the
positive number x where a x
x2 = ab
x = ab
x
=
b
Thm. 7-1
Thm.
If the altitude is drawn from the
vertex of the right angle of a right
Fractals and SelfSimilarity
Section 6-6
iteration - a process of repeating
the same procedure over and over
again.
fractal a geometric figure that is
created using iteration. It is
infinite in structure.
One characteristic of fractals is that
they have se
Coordinate Proof
Section 6-6
Guidelines for Placing Figures on
a Coordinate Plane
1. Use the origin as a vertex or center.
2. Place at least one side of a polygon on
an axis.
3. Keep the figure within the 1st quadrant
if possible.
4. Use coordinates that
Trapezoids and
Kites
Kites
Section 6-5
trapezoid a quadrilateral with
exactly one pair of parallel sides
The parallel sides are called bases.
The nonparallel sides are called legs
base
leg
leg
base
isosceles trapezoid a
trapezoid that has congruent
legs
T
Parts of Similar
Triangles
Triangles
Section 6-5
Thm. 6.7 Proportional
Perimeters
Perimeters
If 2 triangles are similar, then the
perimeters are proportional to the
measures of corresponding sides.
6
P1=24
8
10
3
P
P
1
2
2
=
1
4
5
P2=12
Theorem 6.8
Theore
Special Parallelograms
Section 6-4
Thms. 6-9, 6-10, & 6-11
Each diagonal of a rhombus bisects
two angles of the rhombus.
The diagonals of a rhombus are
perpendicular.
The diagonals of a rectangle are
congruent.
rhombus
rectangle
square
Converses of The
Parallel Lines &
Proportional Parts
Section 6-4
Thm. 6.4 Triangle Proportionality
If a line is parallel to one side of a
triangle and intersects the other 2
sides in 2 distinct points, then it
separates these sides into segments
of proportional lengths.
x
Proving That a
Quadrilateral is a
Parallelogram
Section 6-3
A quadrilateral is a parallelogram if any
one of the following is true.
Both pairs of opposite sides are parallel.
(Definition)
Both pairs of opposite sides are congruent.
(Thm. 6-7)
Both pair
Similar Triangles
Similar
Section 6-3
AA Similarity
AA
If 2 angles of one triangle are
congruent to 2 angles of another
triangle, then the triangles are
similar.
ABC ~ DEF
B
A
E
C
D
F
SSS Similarity
SSS
If the measures of the
corresponding sides of 2 tria
Similar Polygons
Section 6-2
similar figures when figures have
the same shape but are different
sizes
The symbol
~ means is similar to.
Defn. of Similar Polygons
Two polygons are similar iff
if their corresponding angles
are congruent and the
measures of
Properties of
Parallelograms
Parallelograms
Section 6-2
quadrilateral a 4-sided polygon
parallelogram a quadrilateral
with both pairs of opposite sides
parallel
Theorems 6-1, 6-2, 6-3
Theorems
Opposite
sides of a parallelogram are
congruent.
Opposite angl
Proportions
Proportions
Section 6-1
ratio a comparison of 2
quantities
The ratio of a to b can be
expressed as a:b or a
b
where b is not 0.
A special ratio often found in
nature and used by artists and
architects is the golden ratio. It
is approximately 1
Inequalities in Triangles
Inequalities
Section 5-5
Exterior Angle Inequality Thm.
Exterior
The measure of an exterior angle of a
The
triangle is greater than the measure
of each of its remote interior angles.
of
m4 > m1
2
1
34
m4 > m2
Theorem 5-10
Theorem
Inverses,
Contrapositives, and
Indirect Reasoning
Section 5-4
negation the denial of a
statement
Ex. An angle is obtuse.
Negation An angle is not
obtuse.
Inverse ~p
~q
An inverse statement can be formed
by negating both the hypothesis and
conclusion.
If i
The Triangle Inequality Thm.
The
&
Inequalities Involving 2 Triangles
Section 5-4 and 5-5
Triangle Inequality Thm.
Triangle
The sum of the lengths of any two
sides of a triangle is greater than
the length of the 3rd side.
B
6
A
AB + BC > AC
7
10
C
Theorem
Triangle Centers
Section 5-3
Unlike
squares and circles,
triangles have many centers. The
ancient Greeks found four:
incenter, centroid, circumcenter,
and orthocenter.
Centroid
The centroid is formed by the
intersection of the medians of a
triangle.
2x
x
Indirect Proof
Section 5-3
Steps for Writing an Indirect
Proof
1.
2.
3.
Assume the conclusion is false.
Show that this assumption leads to a
contradiction of the hypothesis, or some
other fact, such as a definition, postulate,
theorem, or corollary.
Point