The Central Idea
Parallelism and perpendicularity describe two different relationships between lines.
- Parallel lines lie in the same plane and do not intersect.
- Perpendicular lines intersect to form right angles.
Symbolically, if lines \(\ell\) and \(m\) are parallel, we write
\[ \ell\parallel m. \]If they are perpendicular, we write
\[ \ell\perp m. \]These symbols describe geometric relationships, not arithmetic operations.
Lines in a Plane
In Euclidean plane geometry, two distinct lines have only two possibilities: they intersect at exactly one point, or they do not intersect at all.
If they do not intersect, they are parallel.
\[ \boxed{\text{distinct coplanar lines: intersecting or parallel}} \]This classification depends on the lines being in the same plane.
What Are Parallel Lines?
Two parallel lines are distinct coplanar lines that do not intersect.
If line \(AB\) is parallel to line \(CD\), we may write
\[ \overleftrightarrow{AB}\parallel\overleftrightarrow{CD}. \]The words distinct and coplanar matter. A line is not normally described as parallel to itself in elementary Euclidean geometry, and two nonintersecting lines in different planes need not be parallel.
Parallel Lines Never Meet
Parallel lines extend indefinitely in both directions without sharing a point.
A drawing may make parallel lines appear to approach one another because of perspective, page layout, or imperfect sketching. The geometric definition does not depend on visual appearance.
If two lines are known to be parallel, their nonintersection is a geometric property of the ideal lines themselves.
What Are Perpendicular Lines?
Two lines are perpendicular when they intersect to form a right angle.
If \(\ell\perp m\), then the angle formed at their intersection has measure
\[ 90^\circ=\frac{\pi}{2}. \]Because intersecting lines form four angles, one right angle forces all four angles to be right angles.
Why One Right Angle Is Enough
Suppose lines \(\ell\) and \(m\) intersect. Adjacent angles formed by two intersecting lines are supplementary, so their measures add to \(180^\circ\).
If one angle measures \(90^\circ\), an adjacent angle must measure
\[ 180^\circ-90^\circ=90^\circ. \]The opposite angles have the same measure. Therefore all four angles are right angles.
Right-Angle Marks
Diagrams commonly show perpendicularity with a small square at an intersection. The square represents a right angle.
A right-angle mark is meaningful information supplied by the diagram. By contrast, an unmarked angle that merely looks like \(90^\circ\) is not sufficient evidence of perpendicularity.
Parallel Marks
Parallel lines in a diagram are often marked with matching arrow symbols. One pair of matching arrows may identify one parallel pair; two matching arrows may identify another.
These markings let the diagram communicate relationships without relying on visual scale.
Intersecting Is Not the Same as Perpendicular
All perpendicular lines intersect, but not all intersecting lines are perpendicular.
Two lines may meet at angles such as \(30^\circ\) and \(150^\circ\). Those lines intersect, but they do not form right angles.
\[ \boxed{\text{perpendicular}\Longrightarrow\text{intersecting}} \]The converse is not generally true.
Coincident Lines
Two line descriptions may refer to the same geometric line. Such lines are called coincident.
For example, if points \(A\), \(B\), and \(C\) are collinear, then
\[ \overleftrightarrow{AB}=\overleftrightarrow{AC}. \]Coincident lines share every point. They are therefore different from distinct parallel lines, which share no points.
Skew Lines
In three-dimensional geometry, two lines can fail to intersect without being parallel.
Skew lines are lines that do not intersect and are not coplanar.
This is why the word coplanar belongs in the definition of parallel lines.
\[ \boxed{\text{nonintersecting does not automatically mean parallel in 3D}} \]Euclid's Parallel Postulate
The special role of parallel lines in Euclidean geometry comes from Euclid's fifth postulate. In one modern equivalent form:
Through a point not on a given line, exactly one line can be drawn parallel to the given line.
This statement is often called the Playfair axiom. Within Euclidean geometry, it captures the uniqueness of a parallel through an external point.
Exactly One Parallel Through an External Point
Let \(P\) be a point not on line \(\ell\). Euclidean geometry guarantees one and only one line through \(P\) that is parallel to \(\ell\).
There are two claims inside the phrase exactly one:
- Existence: a parallel line through \(P\) can be drawn.
- Uniqueness: no second distinct parallel through \(P\) exists.
This uniqueness is a defining feature of Euclidean parallelism.
Exactly One Perpendicular Through a Point
Euclidean geometry also gives a useful uniqueness result for perpendicular lines.
Through a given point, there is exactly one line perpendicular to a given line.
The point may lie on the original line or may lie outside it. In either case, the perpendicular through that point is unique.
Perpendicular Through a Point on a Line
Suppose \(P\) lies on line \(\ell\). The unique line through \(P\) that forms a right angle with \(\ell\) is perpendicular to \(\ell\).
This construction is used constantly when building rectangles, coordinate axes, altitudes, perpendicular bisectors, normal lines, and many other geometric objects.
Perpendicular Through an External Point
Suppose \(P\) does not lie on line \(\ell\). A unique perpendicular can be drawn from \(P\) to \(\ell\).
If the perpendicular meets \(\ell\) at \(Q\), then \(Q\) is often called the foot of the perpendicular from \(P\) to \(\ell\).
\[ \overline{PQ}\perp\ell. \]Distance from a Point to a Line
The distance from a point to a line is measured along the perpendicular segment from the point to the line.
If \(P\) is outside line \(\ell\), and \(Q\) is the foot of the perpendicular, then
\[ d(P,\ell)=PQ. \]This perpendicular segment is the shortest segment from \(P\) to any point on \(\ell\).
Why Perpendicular Distance Is the Shortest
Let \(Q\) be the foot of the perpendicular from \(P\) to line \(\ell\), and let \(R\) be any other point on \(\ell\).
Triangle \(PQR\) is a right triangle with \(PR\) as its hypotenuse. The hypotenuse is longer than either leg, so
\[ PR>PQ. \]Therefore \(PQ\) gives the minimum distance from \(P\) to the line.
Distance Between Parallel Lines
The distance between two parallel lines is the length of a perpendicular segment connecting them.
In Euclidean geometry, this perpendicular distance is constant. If \(\ell\parallel m\), then every perpendicular segment joining \(\ell\) to \(m\) has the same length.
This constant separation is one way to understand the geometric uniformity of parallel lines.
Two Lines Perpendicular to the Same Line
In a plane, if two distinct lines are perpendicular to the same line, then they are parallel to each other.
If
\[ \ell\perp n \qquad\text{and}\qquad m\perp n, \]then
\[ \boxed{\ell\parallel m}. \]Both lines have the same right-angle orientation relative to \(n\).
A Line Perpendicular to One of Two Parallels
If two lines are parallel and a third line is perpendicular to one of them, then the third line is perpendicular to the other as well.
If
\[ \ell\parallel m \qquad\text{and}\qquad n\perp\ell, \]then
\[ \boxed{n\perp m}. \]Parallelism Is Transitive
For distinct coplanar lines, parallelism behaves transitively.
If
\[ \ell\parallel m \qquad\text{and}\qquad m\parallel n, \]then
\[ \boxed{\ell\parallel n}. \]Geometrically, all three lines share the same direction in the plane.
Perpendicularity Is Not Transitive
Perpendicularity behaves differently.
If
\[ \ell\perp m \qquad\text{and}\qquad m\perp n, \]then \(\ell\) and \(n\) are generally parallel, not perpendicular.
\[ \boxed{\ell\perp m\text{ and }m\perp n\Longrightarrow\ell\parallel n} \]This is a common place to confuse geometric relationships with transitive equality.
Directions and Angles
Parallel lines have the same direction, allowing for the fact that a line extends in both directions.
Perpendicular lines differ in direction by a quarter-turn:
\[ 90^\circ=\frac{\pi}{2}. \]This directional viewpoint becomes especially useful on the coordinate plane, where direction can be represented by slope.
Parallel Lines on the Coordinate Plane
For nonvertical lines, slope measures direction. Two distinct nonvertical lines are parallel exactly when they have the same slope.
If
\[ y=m_1x+b_1 \qquad\text{and}\qquad y=m_2x+b_2, \]then, for distinct lines,
\[ \boxed{m_1=m_2\Longrightarrow\text{parallel}}. \]The intercepts must be different if the lines are to remain distinct.
Same Slope, Different Intercepts
Consider
\[ y=3x+2 \]and
\[ y=3x-5. \]Both lines have slope \(3\), but their \(y\)-intercepts are different. Therefore,
\[ \boxed{y=3x+2\parallel y=3x-5}. \]Same Slope and Same Intercept
Suppose two equations both simplify to
\[ y=2x+1. \]Then they do not describe two distinct parallel lines. They describe the same line.
Equal slope and equal intercept mean coincident lines.
Vertical Parallel Lines
Vertical lines have equations of the form
\[ x=a. \]Their slopes are undefined, so the ordinary equal-slope test is not applied numerically.
Any two distinct vertical lines are parallel. For example,
\[ \boxed{x=2\parallel x=-4}. \]Horizontal Parallel Lines
Horizontal lines have equations of the form
\[ y=b. \]Every horizontal line has slope \(0\). Therefore any two distinct horizontal lines are parallel.
For example,
\[ \boxed{y=7\parallel y=-1}. \]Perpendicular Slopes
For two nonvertical, nonhorizontal perpendicular lines, their slopes are negative reciprocals.
If one slope is \(m\neq0\), the perpendicular slope is
\[ \boxed{-\frac{1}{m}}. \]Equivalently, if both slopes are defined and nonzero,
\[ \boxed{m_1m_2=-1}. \]What Does Negative Reciprocal Mean?
To find the negative reciprocal of a nonzero number:
- take its reciprocal;
- change its sign.
If
\[ m=\frac{2}{3}, \]then the reciprocal is \(\frac{3}{2}\), and the negative reciprocal is
\[ \boxed{-\frac{3}{2}}. \]Example: Perpendicular Slopes
Suppose one line has slope
\[ m_1=4. \]Write \(4\) as \(\frac{4}{1}\). Its reciprocal is \(\frac{1}{4}\), and changing the sign gives
\[ m_2=-\frac14. \]Check:
\[ 4\left(-\frac14\right)=-1. \]Therefore the directions are perpendicular.
A Negative Slope Can Have a Positive Perpendicular Slope
Suppose
\[ m_1=-\frac{5}{2}. \]Its reciprocal is \(-\frac25\). Changing the sign gives
\[ \boxed{m_2=\frac25}. \]Their product is
\[ -\frac52\cdot\frac25=-1. \]Horizontal and Vertical Lines Are Perpendicular
A horizontal line and a vertical line meet at a right angle.
Thus
\[ y=b \]is perpendicular to
\[ x=a. \]This is the special case that cannot be handled by multiplying two ordinary numerical slopes, because the vertical slope is undefined.
Do Not Say the Vertical Slope Is Infinite
In elementary coordinate geometry, the slope of a vertical line is undefined, not infinity.
The slope formula would require division by zero:
\[ m=\frac{y_2-y_1}{x_2-x_1}. \]On a vertical line, \(x_2-x_1=0\), so the quotient is not a real number.
Testing Parallelism from Two Pairs of Points
Suppose line \(AB\) passes through \(A(1,2)\) and \(B(5,10)\). Its slope is
\[ m_{AB}=\frac{10-2}{5-1}=\frac84=2. \]Suppose line \(CD\) passes through \(C(-2,1)\) and \(D(1,7)\). Its slope is
\[ m_{CD}=\frac{7-1}{1-(-2)}=\frac63=2. \]The slopes are equal. If the lines are distinct, then
\[ \boxed{\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}}. \]Testing Perpendicularity from Two Pairs of Points
Suppose one line has slope
\[ m_1=\frac32 \]and another has slope
\[ m_2=-\frac23. \]Their product is
\[ \frac32\left(-\frac23\right)=-1. \]Therefore the lines are perpendicular.
Finding a Parallel Line Through a Point
Find the equation of the line through \((2,-1)\) parallel to
\[ y=3x+4. \]A parallel line must have slope \(3\). Use point-slope form:
\[ y-(-1)=3(x-2). \]Simplify:
\[ y+1=3x-6, \] \[ \boxed{y=3x-7}. \]Finding a Perpendicular Line Through a Point
Find the equation of the line through \((4,1)\) perpendicular to
\[ y=2x-5. \]The original slope is \(2\), so the perpendicular slope is
\[ -\frac12. \]Use point-slope form:
\[ y-1=-\frac12(x-4). \]Simplify:
\[ y-1=-\frac12x+2, \] \[ \boxed{y=-\frac12x+3}. \]Parallel Lines in Standard Form
Lines do not have to be written in slope-intercept form before their relationship can be recognized.
Consider
\[ 2x+3y=6 \]and
\[ 4x+6y=1. \]Solve each for \(y\):
\[ y=-\frac23x+2 \]and
\[ y=-\frac23x+\frac16. \]The slopes are equal and the lines are distinct, so they are parallel.
A Standard-Form Parallel Test
For nonvertical lines in standard form
\[ A_1x+B_1y=C_1 \]and
\[ A_2x+B_2y=C_2, \]the slopes are \(-A_1/B_1\) and \(-A_2/B_2\). Thus equal direction can be detected when the coefficient pairs \((A_1,B_1)\) and \((A_2,B_2)\) are proportional.
The constants must still be checked to distinguish parallel lines from coincident lines.
A Standard-Form Perpendicular Test
For lines
\[ A_1x+B_1y=C_1 \]and
\[ A_2x+B_2y=C_2, \]a useful perpendicular condition is
\[ \boxed{A_1A_2+B_1B_2=0}. \]This is the dot-product condition for their normal vectors. It also handles horizontal and vertical lines cleanly.
This formula is an advanced shortcut; the slope method remains the main method at this stage.
Example: Standard-Form Perpendicular Lines
Consider
\[ 2x+3y=7 \]and
\[ 3x-2y=5. \]Using the coefficient test,
\[ (2)(3)+(3)(-2)=6-6=0. \]Therefore the lines are perpendicular.
Direction Vectors
Slope is one way to encode line direction. A more general method uses a direction vector.
For a nonvertical line with slope \(m\), one possible direction vector is
\[ \langle 1,m\rangle. \]Parallel lines have proportional direction vectors. Perpendicular direction vectors have dot product zero.
This viewpoint will become increasingly useful in higher-dimensional geometry and linear algebra.
Parallel Direction Vectors
Suppose one line has direction vector
\[ \mathbf{u}=\langle2,3\rangle \]and another has
\[ \mathbf{v}=\langle4,6\rangle. \]Since
\[ \mathbf{v}=2\mathbf{u}, \]the direction vectors are proportional, so the lines have parallel directions.
Perpendicular Direction Vectors
Suppose
\[ \mathbf{u}=\langle2,3\rangle \]and
\[ \mathbf{v}=\langle3,-2\rangle. \]Their dot product is
\[ \mathbf{u}\cdot\mathbf{v}=2(3)+3(-2)=0. \]Therefore the directions are perpendicular.
Coordinate Axes as a Basic Example
The \(x\)-axis and \(y\)-axis provide the standard coordinate example of perpendicular lines.
Their equations are
\[ y=0 \]and
\[ x=0. \]They intersect at the origin and form four right angles.
\[ \boxed{x\text{-axis}\perp y\text{-axis}}. \]Perpendicular Bisectors
A perpendicular bisector of a segment is a line that passes through the segment's midpoint and is perpendicular to the segment.
It combines two conditions:
- the line bisects the segment;
- the line meets the segment at a right angle.
Neither condition alone is enough.
The Perpendicular-Bisector Locus
Every point on the perpendicular bisector of segment \(AB\) is equidistant from \(A\) and \(B\).
If \(P\) lies on the perpendicular bisector, then
\[ \boxed{PA=PB}. \]Conversely, every point equidistant from \(A\) and \(B\) lies on the perpendicular bisector. This makes the perpendicular bisector a geometric locus.
Constructing a Perpendicular Bisector
A classical compass-and-straightedge construction finds the perpendicular bisector of segment \(AB\).
- Open the compass to a radius greater than half of \(AB\).
- Draw arcs centered at \(A\) on both sides of the segment.
- Without changing the radius, draw matching arcs centered at \(B\).
- Connect the two arc-intersection points with a straight line.
The resulting line is perpendicular to \(AB\) and passes through its midpoint.
Constructing a Perpendicular at a Point
If point \(P\) lies on line \(\ell\), a compass can mark two points on \(\ell\) equally distant from \(P\). Constructing the perpendicular bisector of the segment between those two points produces the unique perpendicular through \(P\).
This construction reduces a new problem to a previously understood construction.
Constructing a Parallel
A parallel through an external point can be constructed by several classical methods. One approach constructs a perpendicular to the given line through the external point, then constructs a second perpendicular through that same point to the first perpendicular.
Because two coplanar lines perpendicular to the same line are parallel, the final line is parallel to the original.
Parallel Lines and Rectangles
A rectangle combines both major line relationships.
- Opposite sides are parallel.
- Adjacent sides are perpendicular.
This simple figure illustrates how parallelism and perpendicularity work together to organize larger geometric structures.
Parallel and Perpendicular in Coordinate Design
Coordinate geometry converts geometric relationships into algebraic tests. This allows line relationships to be verified from equations or data rather than from a drawing.
- same slope \(\rightarrow\) parallel direction;
- negative-reciprocal slopes \(\rightarrow\) perpendicular direction;
- horizontal and vertical \(\rightarrow\) perpendicular special case.
The geometry and algebra describe the same underlying structure.
Why the Slope Tests Work
Slope records a line's rise relative to its run. Equal slopes encode equal direction, so distinct lines with equal slopes remain parallel.
A quarter-turn changes a direction vector \(\langle a,b\rangle\) into a perpendicular direction such as
\[ \langle-b,a\rangle. \]If the original slope is \(b/a\), the rotated slope is
\[ \frac{a}{-b}=-\frac{a}{b}, \]which is the negative reciprocal.
Angle Between Two Lines
Parallel and perpendicular lines are two special cases of the more general idea of the angle between lines.
- Parallel directions have angle \(0^\circ\) between them.
- Perpendicular directions have angle \(90^\circ\) between them.
Later courses may compute arbitrary angles between lines using slopes, vectors, dot products, or trigonometric functions.
Common Error: Trusting Appearance
A sketch may show two lines that look parallel or perpendicular even when no relationship has been given.
Do not infer a geometric condition solely from appearance. Use markings, definitions, equations, measured data, constructions, or proved results.
Common Error: Calling Any Nonintersecting Lines Parallel
In the plane, distinct nonintersecting lines are parallel. In three dimensions, nonintersecting lines may instead be skew.
Therefore the complete definition requires the lines to be coplanar.
Common Error: Confusing Equal Slopes with Distinct Parallel Lines
Equal slopes establish the same direction, but two equations may still describe the same line.
Compare
\[ y=2x+3 \]and
\[ 2y=4x+6. \]The second equation simplifies to the first. These lines coincide; they are not two distinct parallel lines.
Common Error: Taking Only the Reciprocal
If a line has slope
\[ \frac34, \]a perpendicular slope is not \(\frac43\). The sign must also change:
\[ \boxed{-\frac43}. \]Check the product:
\[ \frac34\left(-\frac43\right)=-1. \]Common Error: Negating Without Taking the Reciprocal
If the original slope is \(5\), changing only the sign gives \(-5\), which is not perpendicular.
The correct perpendicular slope is
\[ \boxed{-\frac15}. \]Common Error: Applying the Product Rule to a Vertical Line
The rule
\[ m_1m_2=-1 \]assumes both slopes are defined and nonzero. It does not directly apply to a horizontal-vertical pair because the vertical slope is undefined.
Handle horizontal and vertical lines as a special perpendicular case.
Common Error: Assuming Perpendicularity Is Transitive
If \(a\perp b\) and \(b\perp c\), it is tempting to write \(a\perp c\). That is incorrect.
In the same plane, \(a\) and \(c\) have the same direction, so
\[ \boxed{a\parallel c}. \]Common Error: Confusing Segments with Supporting Lines
Two drawn segments may not visibly meet even though their supporting lines would intersect if extended.
Parallelism is fundamentally a relationship between complete lines. When a problem discusses segments, identify the lines containing those segments if necessary.
A Reliable Procedure for Testing Two Lines
- Identify whether the lines are in the same plane.
- Check whether they are vertical, horizontal, or neither.
- If equations are given, put them in a form that reveals direction.
- Find each slope when the slopes are defined.
- Equal slopes indicate parallel direction.
- Negative-reciprocal slopes indicate perpendicular direction.
- Check whether equal-slope equations are actually the same line.
- Use the horizontal-vertical special case when appropriate.
- State the relationship using correct notation.
A Reliable Procedure for Writing a Parallel Line
- Find the slope of the given line.
- Use the same slope for the new line.
- Substitute the required point using point-slope form.
- Simplify if a different equation form is desired.
- Check that the new line is distinct from the original unless coincidence is intended.
A Reliable Procedure for Writing a Perpendicular Line
- Find the slope or direction of the given line.
- If the slope is nonzero and finite, take its negative reciprocal.
- If the original line is horizontal, use a vertical perpendicular.
- If the original line is vertical, use a horizontal perpendicular.
- Use the required point to determine the new equation.
- Verify the right-angle relationship.
Example: Classify Three Lines
Consider
\[ \ell_1:y=\frac12x+3, \] \[ \ell_2:y=\frac12x-4, \]and
\[ \ell_3:y=-2x+1. \]The first two slopes are equal, so
\[ \ell_1\parallel\ell_2. \]Also,
\[ \frac12(-2)=-1, \]so
\[ \ell_1\perp\ell_3 \qquad\text{and}\qquad \ell_2\perp\ell_3. \]Example: Find a Missing Slope
Line \(a\) has slope
\[ -\frac37. \]Find the slope of a line perpendicular to \(a\).
Reverse numerator and denominator:
\[ -\frac73. \]Then change the sign:
\[ \boxed{\frac73}. \]Example: Decide Whether Two Lines Are Parallel
Consider
\[ 3x-2y=8 \]and
\[ 6x-4y=-5. \]Solve the first equation for \(y\):
\[ y=\frac32x-4. \]Solve the second:
\[ y=\frac32x+\frac54. \]The slopes match and the intercepts differ. Therefore,
\[ \boxed{3x-2y=8\parallel6x-4y=-5}. \]Example: Decide Whether Two Lines Are Perpendicular
Consider
\[ y=-\frac43x+2 \]and
\[ 3x-4y=8. \]Rewrite the second equation:
\[ -4y=-3x+8, \] \[ y=\frac34x-2. \]The slopes satisfy
\[ -\frac43\cdot\frac34=-1. \]Therefore the lines are perpendicular.
Example: A Vertical Case
Find the line through \((6,-2)\) perpendicular to
\[ y=5. \]The given line is horizontal. A perpendicular line must be vertical. The vertical line through \((6,-2)\) is
\[ \boxed{x=6}. \]Example: A Horizontal Case
Find the line through \((-3,4)\) perpendicular to
\[ x=8. \]The given line is vertical. A perpendicular line must be horizontal. The horizontal line through \((-3,4)\) is
\[ \boxed{y=4}. \]Geometric Relationships Can Be Proved in Different Languages
The same line relationship may be established in several ways.
- A diagram may explicitly mark parallelism or a right angle.
- A construction may guarantee the relationship.
- A theorem may prove the relationship from earlier facts.
- Coordinate equations may establish it through slope.
- Vectors may establish it through proportionality or a zero dot product.
Mathematics becomes more powerful when we can move between these representations.
Parallelism and Perpendicularity as Constraints
A geometric condition restricts what a figure is allowed to do.
Saying that two lines are parallel fixes their relative direction. Saying that they are perpendicular fixes their directional difference at a quarter-turn.
These constraints reduce the number of possible configurations and make deductive reasoning possible.
Why These Relationships Matter
Parallel and perpendicular relationships support many later ideas, including:
- transversal angle theorems;
- triangle and polygon proofs;
- rectangles, parallelograms, and other quadrilaterals;
- altitudes and perpendicular bisectors;
- coordinate geometry and analytic proofs;
- distance from points to lines and planes;
- orthogonal coordinate systems;
- vectors, dot products, projections, and normal directions;
- engineering drawings, grids, maps, and design constraints.
Practice 1: Identify the Relationship
Two distinct lines lie in the same plane and never intersect. What is their relationship?
Practice 2: Right Angles
Lines \(a\) and \(b\) intersect, and one of the four angles formed measures \(90^\circ\). What can you conclude about \(a\) and \(b\)?
Practice 3: Same Perpendicular
In one plane,
\[ \ell\perp n \qquad\text{and}\qquad m\perp n. \]What relationship holds between \(\ell\) and \(m\), assuming they are distinct?
Practice 4: Parallel Slopes
Determine whether the lines are parallel:
\[ y=-3x+7 \]and
\[ y=-3x-2. \]Practice 5: Perpendicular Slope
Find the slope of a line perpendicular to a line with slope
\[ \frac56. \]Practice 6: Test Perpendicularity
Determine whether lines with slopes
\[ m_1=-\frac25 \qquad\text{and}\qquad m_2=\frac52 \]are perpendicular.
Practice 7: Write a Parallel Line
Find the equation of the line through \((1,4)\) parallel to
\[ y=2x-3. \]Practice 8: Write a Perpendicular Line
Find the equation of the line through \((2,5)\) perpendicular to
\[ y=-\frac13x+4. \]Practice 9: Vertical and Horizontal Lines
Find the line through \((-4,7)\) perpendicular to
\[ x=3. \]Practice 10: Identify the Error
A student says that a line perpendicular to a line of slope \(\frac34\) has slope \(\frac43\). Explain the error and give the correct slope.
Practice 11: Coincident or Parallel?
Classify the relationship between
\[ y=4x-6 \]and
\[ 2y=8x-12. \]Practice 12: Standard Form
Determine whether
\[ 2x+5y=1 \]and
\[ 5x-2y=9 \]are parallel, perpendicular, or neither.
Practice Answers
Practice 1
The lines are distinct, coplanar, and nonintersecting, so they are parallel.
\[ \boxed{\ell\parallel m}. \]Practice 2
If one angle formed by the intersecting lines is a right angle, all four are right angles. Therefore,
\[ \boxed{a\perp b}. \]Practice 3
Two distinct coplanar lines perpendicular to the same line are parallel:
\[ \boxed{\ell\parallel m}. \]Practice 4
Both lines have slope \(-3\), and their intercepts differ. Therefore,
\[ \boxed{y=-3x+7\parallel y=-3x-2}. \]Practice 5
Take the reciprocal and change the sign:
\[ \boxed{-\frac65}. \]Practice 6
Multiply the slopes:
\[ -\frac25\cdot\frac52=-1. \]Therefore the lines are perpendicular.
Practice 7
A parallel line has slope \(2\). Through \((1,4)\):
\[ y-4=2(x-1). \]Therefore,
\[ \boxed{y=2x+2}. \]Practice 8
The given slope is \(-\frac13\), so the perpendicular slope is \(3\). Through \((2,5)\):
\[ y-5=3(x-2). \]Therefore,
\[ \boxed{y=3x-1}. \]Practice 9
The line \(x=3\) is vertical, so a perpendicular line is horizontal. Through \((-4,7)\), the equation is
\[ \boxed{y=7}. \]Practice 10
The student took the reciprocal but did not change the sign. The correct perpendicular slope is
\[ \boxed{-\frac43}. \]Practice 11
Divide the second equation by \(2\):
\[ y=4x-6. \]Both equations describe the same line, so the lines are coincident.
Practice 12
The coefficient test gives
\[ (2)(5)+(5)(-2)=10-10=0. \]Therefore the lines are perpendicular:
\[ \boxed{2x+5y=1\perp5x-2y=9}. \]What to Remember
- Parallel lines are distinct coplanar lines that do not intersect.
- Perpendicular lines intersect to form right angles.
- Coincident lines are the same line; skew lines are noncoplanar and nonintersecting.
- Through an external point, Euclidean geometry gives exactly one parallel to a given line.
- Through a given point, exactly one line is perpendicular to a given line.
- Perpendicular distance gives the shortest distance from a point to a line.
- Two distinct coplanar lines perpendicular to the same line are parallel.
- A line perpendicular to one of two parallel lines is perpendicular to the other.
- Distinct nonvertical lines with equal slopes are parallel.
- Nonvertical, nonhorizontal lines with negative-reciprocal slopes are perpendicular.
- Horizontal and vertical lines are perpendicular.
- A diagram must supply or justify a relationship; appearance alone is not proof.
The central coordinate shortcut is:
\[ \boxed{\text{same slope}\Rightarrow\text{parallel direction},\qquad m_1m_2=-1\Rightarrow\text{perpendicular direction}} \]with horizontal and vertical lines handled as their own special case.
Where We Go Next
Once parallel and perpendicular lines are established, a third line can cut across them and create a network of related angles. Those angle relationships turn line geometry into a powerful system of deductions.
The next lesson, Transversals and Angle Relationships, develops corresponding, alternate, same-side, vertical, supplementary, and related angle structures and shows how they can be used to prove lines parallel or determine unknown angle measures.