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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 non-vertical lines are parallel if and only if their slopes are equal.
If ℓ1 and ℓ2 are two parallel lines and m1 and m2 their respective slopes, then the following statement is true.
ℓ1∥ℓ2⇔m1=m2
The slope of a vertical line is not defined. Therefore, this theorem only applies to non-vertical 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=m2x+b2
(II): x=m2−m1b1−b2
ℓ1∥ℓ2⇒m1=m2
(I): y=mx+b2
(I): LHS−mx=RHS−mx
m1=m2⇒ℓ1∥ℓ2
Both directions of the biconditional statement have been proved.
ℓ1∥ℓ2⇔m1=m2
Consider two lines on the same coordinate plane and their equations in slope-intercept form. Are the lines parallel?
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.
Consider two lines on the same coordinate plane and their equations in slope-intercept form. Are the lines perpendicular?
Substitute (0,4) & (6,0)
Subtract terms
Put minus sign in front of fraction
ba=b/2a/2
x=6, y=4
ca⋅b=ca⋅b
Calculate quotient
LHS−9=RHS−9
Rearrange equation
It can be seen that the lines are perpendicular and that y=23x−5 passes through (6,4), which corresponds to the flower beds.