Angle Theorems & Definitions (Review). Supplementary = 180 ° Complementary = 90°
2.1 Lines and Angles Acute angle – 0 < x < 90 Right angle - 90 Obtuse angle – 90 < x < 180 ...
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Transcript of 2.1 Lines and Angles Acute angle – 0 < x < 90 Right angle - 90 Obtuse angle – 90 < x < 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
• 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
1 23 4
5 67 8
2.1 Lines and Angles
• 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
1 23 4
5 67 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
A D
EB
C F
EF
DE
BC
AB
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 (mA + mB + mC = 180º)
A
BC
2.2 Triangles
• The measure of an exterior angle of a triangle equals the sum of the measures of the 2 non-adjacent interior angles - m1 + m2 = m4
1
2
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:
• Why use Heron’s formula instead of A = ½ bh?
)(perimeter -semi theis s
where))()((
21 cbas
csbsassA
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:
A
C
6
D
F
E
5
4
B10
2.3 Quadrilaterals
Quadrilateral
Parallelogram Trapezoid
Rectangle Rhombus
Square
IsoscelesTrapezoid
2.3 Quadrilaterals
Polygon Area
Square s2
Rectangle l w
Parallelogram b h
Triangle
Trapezoid
bh21
2121 bbh
2.3 Quadrilaterals
Polygon Perimeter
Triangle 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 = 2r 22/7 or 3.14
• Area of a circle –
Note: Just need area and circumference formulas from this section
r
221 )2( rrrA
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=2r2+2rh S=2rh
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=4r2
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
• What is the domain and the range?
xxf )(
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):
• Examples:180rad
52.74)3.1(180
3.1
87.0)50(180
50
rad
radrad
4.1 Angles
• Standard position – always w.r.t. x-axis
θ
4.2 Defining the Trigonometric Functions
• Diagram:
θyr
x
4.2 Defining the Trigonometric Functions
• Definitions:
θyr
x
222
)csc()sec(
)cot()tan(
)cos()sin(
yxr
y
rθ
x
rθ
y
xθ
x
yθ
r
xθ
r
yθ
4.2 Defining the Trigonometric Functions
• Given one function – find others :
θyr
x
.,3
4)tan(,
5
3)cos(
3
9162545
5
4)sin(
2222222
etcx
yθ
r
xθ
x
xxyxr
r
yθ
4.3 Values of the trigonometric functions
• 45-45-90 triangle:– Leg opposite the 45 angle = a
– Leg opposite the 90 angle =
a
a
a2
a2
45
90 45
4.3 Values of the trigonometric functions
• 30-60-90 triangle:– Leg opposite 30 angle = a
– Leg opposite 60 angle =
– Leg opposite 90 angle = 2a
a2a
a390
a3
30
60
4.3 Values of the trigonometric functions
• Common angles for trigonometry
3x
2r1y
3y2r
1x
2r
2x
2y
45
60
30
4.3 Values of the trigonometric functions
31
3)60tan(
2
1)60cos(
2
3)60sin(
12
2)45tan(
2
2)45cos(
2
2)45sin(
3
3
3
1)30tan(
2
3)30cos(
2
1)30sin(
• Some common trig function values:
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:
• Note:
12)21(.sin21.)12sin( 1
)sin(
1)(sin 1
xassamethenotisx
4.3 Values of the trigonometric functions
60)3(tan60)2
1(cos60)
2
3(sin
45)1(tan45)2
2(cos45)
2
2(sin
30)3
3(tan30)
2
3(cos30)
2
1(sin
111
111
111
• Some common inverse trig function values:
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.
• 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).
ac
b C
B
A
4.4 The Right Triangle
• Given the right triangle oriented as follows:
ac
b C
B
A
b
cB
a
cB
b
aB
a
bB
c
aB
c
bB
a
cA
b
cA
a
bA
b
aA
c
bA
c
aA
)csc()sec()cot(
)tan()cos()sin(
)csc()sec()cot(
)tan()cos()sin(
4.4 The Right Triangle
• Example: Given A = 30, a = 2, solve the triangle.
ac
b C
B
A
124163224:
434332
2
3cos
322
3
1tan
603090,90
222222
baccheck
cccc
bA
bbb
aA
BC
4.4 The Right Triangle
• Example: Solve the triangle given: ac
b C
B
A
2318
1836236
454590,90
452
2sin
2
2
6
23sinsin
2
2222222
1
bb
bbbac
BC
AAc
aA
6,23 ca
4.5 Applications of Right Triangles
• No new material – applicationsof the previous section.
ac
b C
B
A