2.1 Lines and Angles Acute angle – 0 < x < 90 Right angle - 90 Obtuse angle – 90 < x < 180 ...

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2.1 Lines and Angles Acute angle – 0 < x < 90 Right angle - 90 Obtuse angle – 90 < x < 180 Straight angle - 180

Transcript of 2.1 Lines and Angles Acute angle – 0 < x < 90 Right angle - 90 Obtuse angle – 90 < x < 180 ...

Page 1: 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

• Acute angle – 0 < x < 90

• Right angle - 90

• Obtuse angle – 90 < x < 180

• Straight angle - 180

Page 2: 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

Page 3: 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

• 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

Page 4: 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

• 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

Page 5: 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

• 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

Page 6: 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

• 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

Page 7: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

2.2 Triangles

• Triangles classified by number of congruent sides

Types of triangles # sides congruent

scalene 0

isosceles 2

equilateral 3

Page 8: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 9: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

2.2 Triangles

• In a triangle, the sum of the interior angle measures is 180º (mA + mB + mC = 180º)

A

BC

Page 10: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 11: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

2.2 Triangles

• Perimeter of triangle = sum of lengths of sides• Area of a triangle = ½ base height

h

b

Page 12: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 13: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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”

Page 14: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 15: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

2.3 Quadrilaterals

Quadrilateral

Parallelogram Trapezoid

Rectangle Rhombus

Square

IsoscelesTrapezoid

Page 16: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

2.3 Quadrilaterals

Polygon Area

Square s2

Rectangle l w

Parallelogram b h

Triangle

Trapezoid

bh21

2121 bbh

Page 17: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 18: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 19: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 20: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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?

Page 21: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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)?

Page 22: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

3.2 More About Functions

• Graph of

• What is the domain and the range?

xxf )(

Page 23: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 24: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.1 Angles

• Acute angle – 0 < x < 90

• Right angle - 90

• Obtuse angle – 90 < x < 180

• Straight angle - 180

Page 25: 2.1 Lines and 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

Page 26: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.1 Angles

• Converting degrees to radians (definition):

• Examples:180rad

52.74)3.1(180

3.1

87.0)50(180

50

rad

radrad

Page 27: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.1 Angles

• Standard position – always w.r.t. x-axis

θ

Page 28: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.2 Defining the Trigonometric Functions

• Diagram:

θyr

x

Page 29: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.2 Defining the Trigonometric Functions

• Definitions:

θyr

x

222

)csc()sec(

)cot()tan(

)cos()sin(

yxr

y

x

y

x

r

r

Page 30: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

r

x

xxyxr

r

Page 31: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 32: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 33: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.3 Values of the trigonometric functions

• Common angles for trigonometry

3x

2r1y

3y2r

1x

2r

2x

2y

45

60

30

Page 34: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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:

Page 35: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 36: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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:

Page 37: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 38: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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(

Page 39: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 40: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

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

Page 41: 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle - 180

4.5 Applications of Right Triangles

• No new material – applicationsof the previous section.

ac

b C

B

A