Lines and Angles - DDTwo

Lines and Angles

Lines Relationship

Parallel Planes

Thursday, April 9, 2020

Lines Relationship

Parallel Lines

Skew Lines

Perpendicular Lines

Two lines are parallel lines if they do not intersect

and are coplanar.

Two lines are skew lines if they do not intersect and

are not coplanar.

Two lines are perpendicular lines if they intersect

and form right angles.

Lines Relationship

Example

Identify lines that are parallel, skew and perpendicular.

A

B C

D

E

F

G H

Parallel Lines

AB and CD

Skew Lines

BC and HE

Perpendicular Lines

BG and HG

Parallel Planes

Example

Identify parallel planes

A

B C

D

E

F

G H

Plane ADE and plane BCH

Plane ABG and plane CDE

Two planes are parallel lines if they do not intersect.

Parallel and Perpendicular Postulate

Parallel Line Postulate

If there is a line and a point not on the line, then

there is exactly one line through the point parallel to

the given line.

Illustrationn

P

There is exactly one line through P parallel to line n

>

>•

Parallel and Perpendicular Postulate

Perpendicular Line Postulate

If there is a line and a point not on the line, then

there is exactly one line through the point

perpendicular to the given line.

Illustration

n

P

There is exactly one line through P perpendicular to

line n

•

Skew, Parallel Lines, Planes, Perpendicular

d. Plane(s) parallel to plane EFG and containing point A

c. Line(s) perpendicular to CD and containing point A

a. Line(s) parallel to CD and containing point A

b. Line(s) skew to CD and containing point A

Think of each segment in the figure as part of a line. Which line(s) or

plane(s) in the figure appear to fit the description?

AB , HG , and EF all appear parallel to CD , but only AB contains

point A.

Both AG and AH is skew to CD and contain point A.

BC, AD , DE , and FC all appear perpendicular to CD , but only AD

contains point A.

Plane ABC appears parallel to plane EFG and contains point A.

Skew, Parallel Lines, Planes, Perpendicular

Try it yourself ! Guided Practice

1. Look at the diagram in Example 1. Name the lines through point H

that appear skew to CD.

ANSWER AH , EH

Lines and Angles

Lesson 3 – 2

Parallel Lines and Transversal Line


Transversal Line

Transversal Line is a line that intersect two or

more coplanar lines at different points

l1

l2

l3

l1

l2

l3

Example: Name the transversal Line

l1Transversal Line : Transversal Line : l1 l2 l3

1. 2.

Angles Formed by Transversal Line

When a transversal line intersect two lines, 8 angles

are formed as shown in the figure below.

1 2

3 4

5 67 8

l1

l2

l3

CORRESPONDING ANGLES:

ALTERNATE EXTERIOR ANGLES:

SAME SIDE/ CONSECUTIVE INTERIOR ANGLES:

ALTERNATE INTERIOR ANGLES:

1 and 5,

1 and 8, 3 and 6,

3 and 5,

2 and 6,

3 and 7, 4 and 8

2 and 7 4 and 5

4 and 6


Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, corresponding or vertical.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1. ) 1 and 2

2. ) 8 and 2

3. ) 5 and 4

4. ) 1 and 8

5. ) 5 and 2

6. ) 1 3and 14

7. ) 9 and 12

8. ) 9 and 14

9. ) 11 and 13

10. ) 15 and 14

corresponding

vertical

alternate interior

alternate exterior

consecutive interior

consecutive interior

alternate exterior

corresponding

vertical

alternate interior

Examples


Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, corresponding or vertical.

Try it yourself !

3. 4. 5.

Corresponding angles Alternate interior anglesAlternate exterior angles

Properties of Parallel Lines

Lesson 3 – 3

Proofs with Parallel Lines

Theorems on Angles Formed by a Transversal



If two parallel lines is cut by a transversal line

then the following theorems hold.

Alternate Interior Angles are Congruent

Alternate Exterior Angles are Congruent

Consecutive Interior Angles are

Supplementary

Corresponding Angles are Congruent

Corresponding Angles Postulate



Find the measure of the angle

1. ) If m1 = 1250, find m2 =_____

2. ) If m8 =1350, find m2 =_____

3. ) If m5 = 870, find m4 =_____

4. ) If m1 = 1180, find m8 = _____

5. ) If m5 = 780, find m2 =_____

6. ) If m2 = 1260, find m4 =____

7. ) If m9 =5x + 8 and m11 = 12x + 2

8. ) If m9 =2x + 7 and m14 = 3x - 13

9. ) If m11= 14x - 56 and m13 = 6x

10. ) If m5=5x + 25 and m2= 3x - 25

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

For #’s 7-10, Solve for x and the angles

1250

1350

870

1180

1020

540

x = 10 m9 = 580 m11 = 1220

x = 20 m9 = 470 m11 = 470

x = 7 m11 = 420 m13 = 420

x = 20 m6 = 1250 m2 = 550

Example


Try it yourself !

1. If m1 = 105°, find m4, m5, and m8. Tell

which postulate or theorem you use in each case.

2. If m 3 = 68° and m 8 = (2x + 4)°, what is the

value of x? Show your steps.

m4 = 105°

m5 = 105°

m8 = 105°

Vertical Angles Congruence

Corresponding Angles

Alternate Exterior Angles

x = 54

Proving Lines Parallel

Lesson 3 – 3

Prove Lines Parallel



Postulate 16: Corresponding Angles Converse

If two lines are cut by a transversal so the

corresponding angles are congruent, then the

lines are parallel.

Illustration1 2

In the figure, 1 2 then line a // line b


Theorem 3.4: Alternate Interior Angles Converse


alternate interior angles are congruent, then the

lines are parallel.

Illustration



Theorem 3.5: Alternate Exterior Angles Converse


alternate exterior angles are congruent, then the

lines are parallel.

Illustration


1

2


Theorem 3.6: Consecutive Interior Angles Converse


consecutive interior angles are supplementary,

then the lines are parallel.

Illustration

In the figure, m1+ m2 = 180 then line a // line b

2

2


Try it yourself !

Can you prove that lines a and b are parallel?

Explain why or why not.

3. 4. 5. m1 + m 2 = 180°

Parallel and Perpendicular Lines

Lesson 3 – 4

Theorems on Parallel and Perpendicular Lines


Theorems on Parallel and Perpendicular

Three Parallel Lines Theorem

If two lines are parallel to the same line then the

lines are parallel to each other.

Illustration If line l 1 is parallel to line l 2,

Then line l 1 is parallel to l 3

line l 3 is parallel to line l 2

l 1>

l 2>

l 3>


Two Lines Perpendicular to a Line Theorem

In a plane, If two lines are perpendicular to the

same line then the lines are parallel to each other.

Illustration If line l 1 is perpendicular to line l 2,

l 1

l 2

line l 3 is perpendicular to line l 2

l 3

>

>

Then line l 1 is parallel to l 3


Perpendicular Transversal Theorem

In a plane, If a line is perpendicular to one of

two parallel line then it is also perpendicular to

the other line.

Illustration If line l 1 is perpendicular to line l 2,

l 1

l 2

line l 1 is parallel to line l 3

l 3>

>

Then line l 3 is perpendicular to l 2

Equations of Lines in the

Coordinate Plane

Lesson 3.5

Slope of Parallel Lines

Slope of Perpendicular Lines

Writing Equation of Lines


Slope of a LINE

Solving for Slope

1. Given a graph:run

risem

• pick a starting point

• do rise by going directly up

or down

• do run by going left or right

to towards exactly at the

second point

•

•

Example 1

m =6

3or 2

Slope of a LINE

•

Example 2• pick a starting point

• do rise by going directly

up or down

• do run by going left or

right to towards exactly at

the second point

m =

•

5

- 4or -5

4

Slope of the Line

2. Given Two Points (x1 , y1) and (x2 , y2)12

12

xx

yym

Example

1. Passing through (-5, -1) and (2, -1)

Solving for Slope

2. Passing through (3, -1) and (7, -5)

m =-1 -

2

-1- - 5

= 0

m =-5 -

7

-1- 3

= -1

x1 y1 x2 y2

x1 y1 x2 y2

Slope of a LINE

3. Given an Equation: y = mx + b

slope y-intercept

Example: Find the slope of the line given the equation

1. y = -4x - 2

2. 3x - 2y = 6

- 2y = -3x + 6

-2 -2 -2y = 3x - 3

2

-3x -3x

m = - 4

m = 3

2

3. -2x + 5y = 10

5y = 2x + 10

5 5 5y = 2x + 2

5

+2x +2x

m = 2

5

Solving for Slope

Sample Problem

Find the Slope given the following

1.

3. Passing through (2, 4) and (-1, 3)

•

•

•2.

4. Passing through (-2, -1) and (1, -3)

•

5. y = 5 x + 7

6. 7x - 5y = 10

Slope and Lines

Negative slope: falls from

left to right, as in line j

Positive slope: rises from

left to right, as in line k

Zero slope (slope of 0):

horizontal, as in line l

Undefined slope: vertical, as

in line n

Equations of Lines in the

Coordinate Plane

Lesson 3.5

Slope of Parallel Lines

Slope of Perpendicular Lines

Writing Equation of Lines


Parallel and Perpendicular Slope

Parallel Lines has equal slope

Perpendicular Lines has negative reciprocal

slope

Example

1. The slope of Line1 is m = -5. Line 1 is parallel to

Line2. What is the slope of Line2?

2. The slope of Line1 is m = -5. Line1 is perpendicular

to Line2. What is the slope of Line2?

Sample Problem

1. The slope of Line1 is m = 3. Line1 is parallel to


3. The slope of Line1 is m = . Line1 is perpendicular


2. The slope of Line1 is m = . Line1 is parallel to


4. The slope of Line1 is m = . Line1 is perpendicular


3

2

3

-1

3

2

Find the slope of line L1 and L2 then tell whether

the given line is perpendicular, parallel, or neither.

1. L1 has (2, 1) and (4, 3), L2 has (1, 2) and (2,1)

2. L1 has (-3, -1) and (1, -3), L2 has (1, 2) and (3,1)

3. L1 has (-1, 1) and (1, 2), L2 has (-1, 1) and (-3,2)

5. L1 equation is 6x + 3y = 6, L2 equation is 2x + y = 3

4. L1 equation is 5x - 2y = 4, L2 equation is 2x + 5 y =10

m1 = 1 m2 = -1 L1 L2

1

2m1 = - L1 L2m2 = -

1

2

neitherm2 = -1

2m1 =

1

2

L1 L2m1 = 5

2m2 = -

2

5

m1 = -2 m2 = -2 L1 L2

Form of Equation of a LINE

Equation of the line has the following form:

Standard form: Ax + By = C

Slope Y-Intercept form:

y = mx + b

Sample Problem

Changing Equation Form of a Line

A. Change the following equation into Slope Y-intercept Form: y = mx +b

1. y = -5 x + 2 2. 4x - 3y = 12

- 3y = -4x + 12

-3 -3 -3

y = 4 x - 43

-4x -4x

3. 2x – y = 2

-y = -2x + 2

y = 2 x - 2

-2x -2x

Sample Problem


B. Change the following equation into Standard Form Ax + By = C

1. y = 2x - 1 2. 3y + 6 = 2x

-2x-2x

-2x + y = -1

2x - y = 1

-2x-2x

-2x + 3y + 6 = 0

- 6 - 6

-2x + 3y = - 6

2x – 3y = 6

3. y = x + 312

12

x12

x

12

x + y = 3 2

x + 2y = 6

Practice Work


A. Change the following equation into Slope Y-intercept Form: y = mx +b

B. Change the following equation into Standard Form Ax + By = C

1. 5x - y = 2

2. 7x - 2y = 6

3. 3x + 5y = 10

4. 5x = 2 - 3y

5. 2x + y = 3

1. 5x = 2 - 3y

2. 3y = 2 – 4x

3. x = 2 + 3y

4. y = 2 – 3x

5. y = x + 213

Writing Equation of a LINE

Equation of the line has the following form:

Standard form: Ax + By = C

Slope Y-Intercept form:

y = mx + b



To write equation of the line use the formula below and

follow the suggested steps:

Suggested Step

1. Look or solve for the slope first.

2. Look for the point (x , y)

3. Plug in the value of step 1 and 2 into y = mx + b and solve

for b

4. Write equation of line either in standard form or slope

y-intercept form as required.

Slope Intercept

Formula:

y2 – y1

x2 – x1

=mSlope

Formula:y = mx + b

Example:

Given a slope and point, write the equation in standard form Ax + By = C and

slope y-intercept form y = mx + b. Use the formula above:

1. m = 1 and (-2 , 1) x , y

y = mx + b

1 =

Slope Y-Int Form : y = 1x + 3

Standard Form : x – y = -3

2. m = and (-4 , 1) 32 x , y

Slope Y-Int Form: y = + 73x

2

Standard Form : 3x – 2y = -14

STEPS

1. Look/Solve for slope m

2. Look for point (x,y)

3. Plug into y = mx + b

and solve for b.

4. Write equation in two

forms as required.

1-2 + b

1 = -2 + b+2+2

3 = b

y = mx + b

1 = 3/2-4 + b

1 = -6 + b+6+6

7 = b

Slope Intercept Formula: y = mx + by2 – y1

x2 – x1

=mSlope

Formula:

Given two points, write the equation in standard form Ax + By = C and slope y-

intercept form y = mx + b. Use the formula above:

3. (-2 , 5) and (-4, 6)x , y

Standard Form : x + 2y = 8

Solve for slope m:-12

m =

Slope Y-Int Form : y = x + 4-1

2

Slope Intercept Formula: y = mx + by2 – y1

x2 – x1

=mSlope

Formula:

y = mx + b

5 = -1/2 -2 + b

5 = 1 + b- 1- 1

4 = b

STEPS



3. Plug into y = mx + b

and solve for b.


forms as required.

4. (5, -2) and (3 , 6) x , y

Solve for slope m: -4m =

y = mx + b

-2 = -45 + b

-2 = -20 + b+20+20

18 = b

Standard Form : 4x + y = 18

Slope Y-Int Form: y = -4x + 18

Example:

Writing Equation with Parallel and Perpendicular Lines

1. (-1 , 3) m = -8x

STEPS



3. Plug-in, simplify.


forms as required.

y

=y -8x – 5

y = mx + b

5.) Through point (-1, 3) and parallel to 8x + y = 3

3 = -8-1+ b

3 = 8 + b-8-8

-5 = b

Writing Equation with Parallel and Perpendicular Lines

x

STEPS



3. Plug-in, and simplify.


forms as required.

y

y = mx + b

2. (2 , 4) m =-1

2

=y-1x

2+ 5

6) Through point (2, 4) and perpendicular to 2x - y = 3

4 = -1/22 + b

4 = -1 + b+1+1

5 = b

Perpendicular


Line n 2 is parallel to n 1 and passes through the point (3, 2).

Write an equation of n 2.

Find the slope of n 2.

1

3–The slope of n 1 is .

Because parallel lines have the same

slope, the slope of n 2 is also .1

3–

SOLUTION

Use (x1 , y1) = (3, 2) and m = - . 3

1

Line n1 has the equation .1

3– x – 1 y =

Write an equation.

Slope - Y intercept form :3

1y = - x + 3.

Standard Equation : x + 3y = 9

Equation of a LINE

Line r has the equation 2x + 3y = 6. Find the equation of line t perpendicular

to line r that passes through the point ( -1, 2)

SOLUTION

Find the slope of line t

To find the slope of line t, first find slope of line r

Use the equation and get y by itself to find slope

of line r 2x + 3y = 6

3y = -2x + 6

y =3

2 x + 2

Slope m of line rSlope m of line t is m =2

3

Slope of a Line

Horizontal Line

H

O

Y

Horizontal Line

Zero Slope

y = number

is the equation

Slope of a Line

Vertical Line

Ver

No

X

Vertical Line

No Slope

x = number

is the equation

“HOY VerNoX”

Sample Problem

Use “HOY VerNoX”. For each number, give either the slope, equation or the type of line.

1.) y = - 4

2.) x = 7

3.) m = 0 and passing thru the point ( 2, -3)

4.) m = undefined and passing thru the point

( 2, 3)

m = 0 Horizontal line

m = undefined Vertical line

Horizontal line y = -3

Vertical line x = 2

Lines and Angles - DDTwo

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