Transcript 2.1 Lines and Angles • Acute angle – 0 • Right angle - 90 • Obtuse angle – 90 • Straight angle - 180

2.1 Lines and Angles
• Acute angle –
0 < x < 90
• Right angle - 90
• Obtuse angle –
90 < x < 180
• Straight angle - 180
2.1 Lines and Angles
• Complementary
angles – add up to 90
• Supplementary angles
– add up to 180
• Vertical angles – the
angles opposite each
other are congruent
2.1 Lines and Angles
• Intersection – 2 lines
intersect if they have one
point in common.
• Perpendicular – 2 lines are
perpendicular if they
intersect and form right
angles
• Parallel – 2 lines are parallel
if they are in the same plane
but do not intersect
2.1 Lines and Angles
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following congruent pairs of angles are formed:
– Corresponding angles:1 & 5, 2 & 6, 3
& 7, 4 & 8
– Alternate interior angles: 4 & 5, 3 & 6
– Alternate exterior angles: 1 & 8, 2 & 7
2.1 Lines and Angles
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following supplementary pairs of angles are
formed:
– Same side interior angles: 3 & 5, 4 & 6
– Same side exterior angles: 1 & 7, 2 & 8
2.1 Lines and Angles
• When 3 or more parallel lines are cut by a pair of
transversals, the transversals are divided
proportionally by the parallel lines
AB
DE

BC
EF
A
B
C
D
E
F
2.2 Triangles
• Triangles classified by number of congruent sides
Types of triangles
# sides congruent
scalene
0
isosceles
2
equilateral
3
2.4 The Angles of a Triangle
• Triangles classified by angles
Types of triangles
Angles
acute
All angles acute
obtuse
One obtuse angle
right
One right angle
equiangular
All angles congruent
2.2 Triangles
• In a triangle, the sum of the interior angle
measures is 180º
(mA + mB + mC = 180º)
A
C
B
2.2 Triangles
• The measure of an exterior angle of a triangle
equals the sum of the measures of the 2 nonadjacent interior angles - m1 + m2 = m4
2
1
3
4
2.2 Triangles
• Perimeter of triangle = sum of lengths of sides
• Area of a triangle = ½ base  height
h
b
2.2 Triangles
• Heron’s formula – If 3 sides of a triangle
have lengths a, b, and c, then the area A of a
triangle is given by:
A  s ( s  a)(s  b)(s  c) where
s is thesemi - perimeters  12 (a  b  c)
• Why use Heron’s formula instead of
A = ½ bh?
2.2 Triangles
•
Definition: Two Triangles are similar 
two conditions are satisfied:
1. All corresponding pairs of angles are
congruent.
2. All corresponding pairs of sides are
proportional.
Note: “~” is read “is similar to”
2.2 Triangles
•
Given ABC ~ DEF with the following
measures, find the lengths DF and EF:
E
10
5
A
B
D
6
4
C
F
2.3 Quadrilaterals
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Isosceles
Trapezoid
2.3 Quadrilaterals
Polygon
Area
Square
s2
Rectangle
lw
Parallelogram
bh
Triangle
1
2
bh
Trapezoid
1
2
hb1  b2 
2.3 Quadrilaterals
Polygon
Triangle
Perimeter
a + b + c (3 sides)
Quadrilateral
a + b + c + d (4 sides)
Parallelogram
2a + 2b
Rectangle
2l + 2w
Square
4s
2.4 Circumference and Area of a Circle
• Circumference of a circle:
C = d = 2r
  22/7 or 3.14
r
• Area of a circle – A  12 r (2 r )   r 2
Note: Just need area and circumference
formulas from this section
2.6 Solid Geometric Figures
V = volume
A = total surface
area
S = lateral surface
area
Rectangular solid
V=lwh
A=2lw+2lh+2wh
Cube
V=e3
A=6s2
Right circular
cylinder
V=r2h
A=2r2+2rh
S=2rh
Right prism
V=Bh
A=2B+ph
S=ph
Right circular cone V=(1/3) r2h
A=r2+rs
S=rs
Regular pyramid
V=(1/3)Bh
A=B+(1/2)ps
S=(1/2)ps
Sphere
V=(4/3) r3
A=4r2
3.2 More About Functions
Domain:
x-values
(input)
Range:
y-values
(output)
Example: Demand for a product depends on its
price.
Question: If a price could produce more than one
demand would the relation be useful?
3.2 More About Functions
• Function notation:
y = f(x) – read “y equals f of x”
note: this is not “f times x”
• Linear function: f(x) = mx + b
Example: f(x) = 5x + 3
• What is f(2)?
3.2 More About Functions
• Graph of
f ( x)  x
• What is the domain and the range?
3.2 More About Functions - Determining
Whether a Relation or Graph is a Function
• A relation is a function if:
for each x-value there is exactly one y-value
– Function: {(1, 1), (3, 9), (5, 25)}
– Not a function: {(1, 1), (1, 2), (1, 3)}
• Vertical Line Test – if any vertical line
intersects the graph in more than one point,
then the graph does not represent a function
4.1 Angles
• Acute angle –
0 < x < 90
• Right angle - 90
• Obtuse angle –
90 < x < 180
• Straight angle - 180
4.1 Angles
• 45 angle
• Also 360-45 = 315
• 135 angle
• Also 360-135 = 225
4.1 Angles
• Converting degrees to radians (definition):
 rad 180
• Examples:
50 

(50) rad  0.87 rad
180
180
1.3 rad 
(1.3)  74.52

4.1 Angles
• Standard position – always w.r.t. x-axis
θ
4.2 Defining the Trigonometric Functions
• Diagram:
r
y
θ
x
4.2 Defining the Trigonometric Functions
• Definitions:
y
sin(θ ) 
r
y
tan(θ ) 
x
r
sec(θ ) 
x
r 2  x2  y2
x
cos(θ ) 
r
x
cot(θ ) 
y
r
csc(θ ) 
y
r
y
θ
x
4.2 Defining the Trigonometric Functions
• Given one function – find others :
r
4 y
θ
sin(θ )    
5 r 
x
r 2  x 2  y 2  52  x 2  42  x 2  25  16  9
x3
x 3
y 4
cos(θ )   , tan(θ )   , etc.
r 5
x 3
y
4.3 Values of the trigonometric functions
• 45-45-90 triangle:
– Leg opposite the 45 angle = a
– Leg opposite the 90 angle = 2a
45
a
2a
90
45
a
4.3 Values of the trigonometric functions
• 30-60-90 triangle:
– Leg opposite 30 angle = a
– Leg opposite 60 angle = 3a
– Leg opposite 90 angle = 2a
60
2a
30
a
90
3a
4.3 Values of the trigonometric functions
• Common angles for trigonometry
r2
r2
  60
y 3
y 1
  30
x 3
r2
x 1
  45
x 2
y 2
4.3 Values of the trigonometric functions
• Some common trig function values:
1
sin(30) 
2
2
sin(45) 
2
3
sin(60) 
2
3
cos(30) 
2
2
cos(45) 
2
1
cos(60) 
2
1
3
tan(30) 

3
3
2
tan(45) 
1
2
3
tan(60) 
 3
1
4.3 Values of the trigonometric functions
• The inverse trigonometric functions are
defined as the angle giving the result for the
given function (sin, cos, tan, etc.)
• Example: sin(12)  .21 sin 1 (.21)  12
• Note:
1
sin ( x) is not the sam e as
sin(x)
1
4.3 Values of the trigonometric functions
• Some common inverse trig function values:
1
3
3
1
1
sin ( )  30 cos ( )  30 tan ( )  30
2
2
3
2
2
1
1
sin ( )  45 cos ( )  45 tan1 (1)  45
2
2
3
1
1 1
sin ( )  60 cos ( )  60 tan1 ( 3 )  60
2
2
1
4.4 The Right Triangle
• Solving a triangle: Given 3 parts of a triangle (at least
one being a side), we are to find the other 3 parts.
B
c
A
b
a
C
• Solving a right triangle: Since one angle is 90, we
need to know another angle (the third angle will be the
complement) and a side or we need to know 2 of 3
sides (use the Pythagorean theorem to find 3rd side).
4.4 The Right Triangle
• Given the right triangle oriented as follows:
a
sin( A) 
c
b
cot(A) 
a
b
sin(B ) 
c
a
cot(B) 
b
b
cos(A) 
c
c
sec( A) 
b
a
cos(B ) 
c
c
sec(B ) 
a
a
tan(A) 
b
c
csc( A) 
a
b
tan(B ) 
a
c
csc(B ) 
b
B
c
A
b
a
C
4.4 The Right Triangle
• Example: Given A = 30, a = 2, solve the triangle.
B
C  90,
c
B  90  30  60
t an A 
a
1
2

 b2 3
b
3 b
cos A 
b
3 2 3



c
2
c
check :
A
3c  4 3  c  4

c  a b  4  2  2 3
2
2
2
2
b
2

2
 16  4  12
a
C
4.4 The Right Triangle
B
• Example: Solve the triangle given:
c
a  3 2, c  6
A
b
a
3 2
2
2
 sin A 

 A  sin 1
 45
c
6
2
2
C  90, B  90  45  45
sin A 

c  a b  6  3 2
2
2
2
2
b 2  18  b  3 2

2
 b 2  36  18  b 2
a
C
4.5 Applications of Right Triangles
B
• No new material – applications
of the previous section.
c
A
b
a
C