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Here are a few recommended readings before getting started with this lesson.
Two coplanar lines — lines that are on the same plane — that do not intersect are said to be parallel lines. In a diagram, triangular hatch marks are drawn on lines to denote that they are parallel. The symbol $∥$
is used to algebraically denote that two lines are parallel. In the diagram, lines $m$ and $ℓ$ are parallel.
In a coordinate plane, two distinct nonvertical lines are parallel if and only if their slopes are equal.
If $ℓ_{1}$ and $ℓ_{2}$ are two parallel lines and $m_{1}$ and $m_{2}$ their respective slopes, then the following statement is true.
$ℓ_{1}∥ℓ_{2}⇔m_{1}=m_{2}$
The slope of a vertical line is not defined. Therefore, this theorem only applies to nonvertical lines. However, any two distinct vertical lines are parallel.
Since the theorem consists of a biconditional statement, the proof consists of two parts.
$(I):$ $y=m_{2}x+b_{2}$
$(II):$ $x=m_{2}−m_{1}b_{1}−b_{2} $
$ℓ_{1}∥ℓ_{2}⇒m_{1}=m_{2}$
$(I):$ $y=mx+b_{2}$
$(I):$ $LHS−mx=RHS−mx$
$m_{1}=m_{2}⇒ℓ_{1}∥ℓ_{2}$
Both directions of the biconditional statement have been proved.
$ℓ_{1}∥ℓ_{2}⇔m_{1}=m_{2}$
Consider two lines on the same coordinate plane and their equations in slopeintercept form. Are the lines parallel?
Write an equation in slopeintercept form of the line that passes through $(2,3)$ and is parallel to $y=2x−1.$
What do parallel lines have in common?
As seen above, the graph of $y=2x+7$ passes through $(2,3)$ and is parallel to the graph of $y=2x−1.$
Find the slope of the line shown on the graph.
Substitute $(0,3)$ & $(1,1)$
Subtract terms
$1a =a$
As seen above, the graph of $y=2x−4.5$ is parallel to the pavement and passes through $(4,3.5).$
Zosia wants to propose a new mural to be painted on the side of the planetarium. She starts with a moon and two stars that are already painted on the building.
Zosia wants to place more stars in the line that connects the two existing stars. She also wants to make a second line of stars that is parallel to the first and passes through the moon. The two stars and the moon can be represented on a coordinate plane.
Write the equation of the line of stars that passes through the moon. Give the answer in slopeintercept form.Find the slope of the line that passes through the stars.
Substitute $(4,3)$ & $(3,0.5)$
$a−(b)=a+b$
Add terms
$ba =b⋅2a⋅2 $
$ba =b/7a/7 $
$x=2$, $y=1$
$b1 ⋅a=ba $
Put minus sign in front of fraction
$aa =1$
$LHS+1=RHS+1$
Rearrange equation
Two coplanar lines — lines that are on the same plane — that intersect at a right angle are said to be perpendicular lines. The symbol $⊥$
is used to algebraically denote that two lines are perpendicular. In the diagram, lines $m$ and $ℓ$ are perpendicular.
In a coordinate plane, two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals.
If $ℓ_{1}$ and $ℓ_{2}$ are two perpendicular lines and $m_{1}$ and $m_{2}$ their respective slopes, the following relation holds true.
$ℓ_{1}⊥ℓ_{2}⇔m_{1}⋅m_{2}=1$
This theorem does not apply to vertical lines because their slope is undefined. However, vertical lines are always perpendicular to horizontal lines.
Since the theorem is a biconditional statement, the proof consists of two parts.
Side  Points  $Distance Formula(x_{2}−x_{1})_{2}+(y_{2}−y_{1})_{2} $

Length 

$AO$  $A(1,m_{1})$ $&$ $O(0,0)$  $(0−1)_{2}+(0−m_{1})_{2} $  $1+m_{1} $ 
$CO$  $C(1,m_{2})$ $&$ $O(0,0)$  $(0−0)_{2}+(0−m_{2})_{2} $  $1+m_{2} $ 
$CA$  $C(1,m_{2})$ $&$ $A(1,m_{1})$  $(1−1)_{2}+(m_{1}−m_{2})_{2} $  $m_{1}−m_{2}$ 
Substitute expressions
$(a )_{2}=a$
Add terms
$(a−b)_{2}=a_{2}−2ab+b_{2}$
$LHS−m_{1}$