lines and angles - ddtwo
TRANSCRIPT
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
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
Angles Formed by Transversal Line
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
Angles Formed by Transversal Line
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
Thursday, April 9, 2020
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
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Theorems on Angles Formed by a Transversal
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
Theorems on Angles Formed by a Transversal
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
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
Prove Lines Parallel
Theorem 3.4: Alternate Interior Angles Converse
If two lines are cut by a transversal so the
alternate interior angles are congruent, then the
lines are parallel.
Illustration
In the figure, 1 2 then line a // line b
Prove Lines Parallel
Theorem 3.5: Alternate Exterior Angles Converse
If two lines are cut by a transversal so the
alternate exterior angles are congruent, then the
lines are parallel.
Illustration
In the figure, 1 2 then line a // line b
1
2
Prove Lines Parallel
Theorem 3.6: Consecutive Interior Angles Converse
If two lines are cut by a transversal so the
consecutive interior angles are supplementary,
then the lines are parallel.
Illustration
In the figure, m1+ m2 = 180 then line a // line b
2
2
Theorems on Angles Formed by a Transversal
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
Thursday, April 9, 2020
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>
Theorems on Parallel and Perpendicular
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
Theorems on Parallel and Perpendicular
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
Thursday, April 9, 2020
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
Thursday, April 9, 2020
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
Line2. What is the slope of Line2?
3. The slope of Line1 is m = . Line1 is perpendicular
to Line2. What is the slope of Line2?
2. The slope of Line1 is m = . Line1 is parallel to
Line2. What is the slope of Line2?
4. The slope of Line1 is m = . Line1 is perpendicular
to Line2. What is the slope of Line2?
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
Changing Equation Form of a Line
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
Changing Equation Form of a Line
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
Thursday, April 9, 2020
Writing Equation of a LINE
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
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.
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
1. Look/Solve for slope m
2. Look for point (x,y)
3. Plug-in, simplify.
4. Write equation in two
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
1. Look/Solve for slope m
2. Look for point (x,y)
3. Plug-in, and simplify.
4. Write equation in two
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
Writing Equation of a LINE
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
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