Here are a few recommended readings before getting started with this lesson.
The lines $ℓ_{1}$ and $ℓ_{2}$ are positioned as shown in the graph. Move the point C vertically.
In the previous graph, the slope triangles △ABF and △CDE can be mapped onto each other by a translation. Since translations are rigid motions, it can be concluded that △ABF and △CDE are congruent triangles. Because the slope triangles are congruent, the slopes of the lines are equal. This means that the lines are parallel.
In a coordinate plane, two distinct nonvertical lines are parallel if and only if their slopes are equal.
Let $ℓ_{1}$ and $ℓ_{2}$ be parallel lines, and m1 and m2 be their respective slopes. Then, the following statement is true.
The slope of a vertical line is not defined. Therefore, this theorem only applies to nonvertical lines. Any two distinct vertical lines are parallel.
Since the theorem consists of a biconditional statement, the proof will consists in two parts.
$(I):$ y=m2x+b2
$(II):$ $x=m_{2}−m_{1}b_{1}−b_{2} $
$(I):$ y=mx+b2
$(I):$ LHS−mx=RHS−mx
Both directions of the biconditional statement have been proved.
If the equation of a linear function is written in slopeintercept form, its slope can be identified.
By rewriting the given equation in slopeintercept form, find the slope of a parallel line to the line whose equation is shown. If necessary, round your answer to 2 decimal places.
Start by identifying the slope of the given line. Then, use the Slopes of Parallel Lines Theorem.
Up to now, it has been discussed how to find the equation of a parallel line to a line whose equation is given. What about finding the equation of a parallel line to a line whose graph is given?
The first transatlantic telegraph cable was laid between Valentia in western Ireland and Trinity Bay Newfoundland in the 1850s. With the invention of fiber optic cables, the number of transatlantic cables has increased. The following map shows some of these cables.
Kevin wants to write the equation of a line parallel to the first transatlantic cable and passes through point V(4,3), a specific location in Virginia beach. Kevin draws a coordinate plane, then line $ℓ_{1},$ on which the cable lies, and the point V as shown.By observing the graph, it can be seen that the line $ℓ_{1}$ passes through the points (0,3) and (4,4). When a slope triangle is constructed between these points, the slope of the line is calculated as $41 .$
It can be seen that the line intercepts the yaxis at (0,3). Therefore, the yintercept is 3. Knowing the slope and the yintercept of line is enough to write its equation in slopeintercept form.$x=4$, $y=3$
$b1 ⋅a=ba $
Put minus sign in front of fraction
$aa =1$
LHS+1=RHS+1
Rearrange equation
Recall that a system of linear equations can have infinitely many solutions, one solution or no solution. Under which conditions does a system of linear equations have infinitely many solutions or no solution?
Consider the following systems of equations.System I: Infinitely many solutions
System II: No solution
Determine whether the equations in the system represent parallel lines. What does this say about the number of solutions?
Equation  Slope  yIntercept 

$y=5x+(3)$  $5$  $3$ 
$y=5x+(1)$  $5$  $1$ 
By the Slopes of Parallel Lines Theorem, these lines are parallel and do not intersect. Therefore, System (II) has no solution.
From the previous example, the following conclusions about systems of linear equations can be made.
Condition  Conclusion  Example 

The lines of the system have the same slope and the same yintercept.  The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.  
The lines of the system have the same slope but different yintercept.  The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.  
The lines of the system have different slopes.  The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.  
These conclusions can be seen in the following diagram.
In the previous graph, it can be seen that the initial angle between the lines measures $90_{∘}.$ Therefore, the lines 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 with slopes m1 and m2, respectively, the following relation holds true.
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, it will be proven in two parts.
Side  Points  $Distance Formula((x_{2}−x_{1})_{2}+(y_{2}−y_{1})_{2} ) $

Length 

AO  A(1,m1) $&$ O(0,0)  $(0−1)_{2}+(0−m_{1})_{2} $  $1+m_{1} $ 
CO  C(1,m2) $&$ O(0,0)  $(0−0)_{2}+(0−m_{2})_{2} $  $1+m_{2} $ 
CA  C(1,m2) $&$ A(1,m1)  $(1−1)_{2}+(m_{1}−m_{2})_{2} $  m1−m2 
Substitute expressions
$(a )_{2}=a$
Add terms
(a−b)2=a2−2ab+b2
$LHS−m_{1}=RHS−m_{1}$
$LHS−m_{2}=RHS−m_{2}$
$LHS/(2)=RHS/(2)$
Put minus sign in front of fraction
$aa =1$
Rearrange equation
Substitute expressions
$(a )_{2}=a$
Add terms
(a−b)2=a2−2ab+b2
$LHS−m_{1}=RHS−m_{1}$
$LHS−m_{2}=RHS−m_{2}$
$m_{1}m_{2}=1$
a(b)=a⋅b
The biconditional statement has been proven.
If the equation of a linear function is written in slopeintercept form, its slope can be easily identified.
Find the slope of the line perpendicular to the given equation's line. This can be done by rewriting the given equation in slopeintercept form. If necessary, round the answer to two decimal places.
Just like with parallel lines, there is an infinite number of perpendicular lines to a given line. Two pieces of information will help in writing the equation of a perpendicular line. These are the slope — which is calculated using the slope of the given line — and a point that lies on the line.
The equation of a line is given in standard form.Start by identifying the slope of the given line. Then, use the Slopes of Perpendicular Lines Theorem.
It has been discussed how to find the equation of a line that is perpendicular to a line whose equation is given. What about finding the equation of a line perpendicular to a line whose graph is given?
Mark and Paulina have been asked to write an equation of a line perpendicular to the line shown in the graph. Additionally, they are told that this perpendicular line should pass through the point $(0,38 ).$The theorems seen in this lesson can be used to identify quadrilaterals and some of their properties.
Determine whether quadrilateral ABCD is a parallelogram. Explain the reasoning.
Yes, see solution.
Recall that a parallelogram is a quadrilateral with two pairs of parallel sides.
Yes, see solution.
Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.