Parallel & Perpendicular Lines

Study Sheet

Parallel & Perpendicular Lines

Everything you need to name angles, find them, and work with slopes

Kinds of Lines: Parallel, Perpendicular, and Skew

Concept
The three relationships
1234

Parallel lines lie in the same flat surface (plane) and never meet, no matter how far you stretch them. We write ℓ∥m\ell \parallel m. Perpendicular lines cross to form a perfect square corner, a 90∘90^\circ angle. We write ℓ⊥m\ell \perp m. Skew lines do not lie in the same plane. They never meet, but they are not parallel either --- think of an airplane's path crossing high above a straight road.

Two parallel lines cut by a transversal.

Concept
Transversal

A transversal is a line that crosses two (or more) other lines. When a transversal cuts two lines it creates eight angles --- four at each crossing --- and those angles come in special matching pairs.

Tip

Quick check: Parallel lines stay the exact same distance apart forever. Perpendicular is a special case of intersecting --- the angle just happens to be exactly 90∘90^\circ.

The Eight Angles and Their Pairs

When a transversal tt crosses lines mm and nn, number the angles like this.

An angle measures a turn.

Concept
Name the four pairs

Corresponding angles sit in the same corner at each crossing (one up, one down). Pairs: (1,5), (2,6), (3,7), (4,8)(1,5),\ (2,6),\ (3,7),\ (4,8). Alternate interior angles are between the two lines and on opposite sides of the transversal. Pairs: (3,6), (4,5)(3,6),\ (4,5). Alternate exterior angles are outside the two lines and on opposite sides of the transversal. Pairs: (1,8), (2,7)(1,8),\ (2,7). Same-side interior angles (also called co-interior or consecutive interior) are between the lines and on the same side of the transversal. Pairs: (3,5), (4,6)(3,5),\ (4,6).

Tip

Memory helper: Interior means inside (between the two lines: angles 3, 4, 5, 6). Exterior means outside (angles 1, 2, 7, 8). Alternate always means opposite sides of the transversal; same-side means the same side.

Example
Naming a pair

Look at the diagram above. What is the relationship between angle 44 and angle 55? Both angles are between lines mm and nn (interior), and angle 44 is on the right of tt while angle 55 is on the left (opposite sides). They are alternate interior angles.

Angle Relationships When the Lines Are Parallel

Everything above works for any two lines. The magic happens when the two lines are parallel.

An angle measures a turn.

Concept
The parallel-line angle rules

If two parallel lines are cut by a transversal, then:

  • [leftmargin=6mm,itemsep=1pt]
  • Corresponding angles are congruent (equal).
  • Alternate interior angles are congruent.
  • Alternate exterior angles are congruent.
  • Same-side interior angles are supplementary (they add to 180∘180^\circ).

Also, vertical angles (the ones directly across an X) are always congruent, and a straight line gives a linear pair that adds to 180∘180^\circ.

Example
One angle unlocks them all

In the figure, m∥nm \parallel n and angle 1=70∘1 = 70^\circ. Find every other angle.

Angle 1=70∘1=70^\circ. Its vertical angle is 44, so angle 4=70∘4 = 70^\circ. Angles 22 and 33 form linear pairs with angle 11, so each is 180∘−70∘=110∘180^\circ - 70^\circ = 110^\circ. Now jump to the bottom crossing using the parallel rules: angle 55 corresponds to angle 11, so angle 5=70∘5 = 70^\circ. Then angle 8=70∘8 = 70^\circ (vertical to 55), and angles 66 and 77 are 110∘110^\circ each. Result: angles 1,4,5,8=70∘1,4,5,8 = 70^\circ and angles 2,3,6,7=110∘2,3,6,7 = 110^\circ.

Example
Same-side interior are supplementary

Suppose m∥nm \parallel n and one same-side interior angle measures 115∘115^\circ. Its partner satisfies 115∘+x=180∘115^\circ + x = 180^\circ, so x=65∘x = 65^\circ.

Tip

Shortcut: With parallel lines, every one of the eight angles is either equal to the angle you know or is its supplement (180∘180^\circ minus it). There are really only two different numbers in the whole picture!

Finding Unknown Angles with Algebra

Concept
Turn a relationship into an equation
55°

Decide whether the two labeled angles are congruent (equal) or supplementary (add to 180∘180^\circ). Then write the matching equation and solve for the variable. Finally, plug back in to find the actual angle measures.

An angle measures a turn.

Example
Congruent angles (equal)

Lines m∥nm \parallel n. Two corresponding angles measure (3x+15)∘(3x+15)^\circ and (5x−25)∘(5x-25)^\circ. Find xx and each angle.

Corresponding angles are congruent, so set them equal:

3x+15=5x−2540=2xx=20\begin{aligned} 3x+15 &= 5x-25\\ 40 &= 2x\\ x &= 20 \end{aligned}

Each angle is 3(20)+15=75∘3(20)+15 = 75^\circ (and 5(20)−25=75∘5(20)-25 = 75^\circ as a check). x=20x=20, both angles =75∘=75^\circ.

Example
Supplementary angles (add to 180)

Lines m∥nm \parallel n. Two same-side interior angles measure (2x+20)∘(2x+20)^\circ and (3x−10)∘(3x-10)^\circ. Find xx and each angle.

(2x+20)+(3x−10)=1805x+10=1805x=170x=34\begin{aligned} (2x+20)+(3x-10) &= 180\\ 5x+10 &= 180\\ 5x &= 170\\ x &= 34 \end{aligned}

The angles are 2(34)+20=88∘2(34)+20 = 88^\circ and 3(34)−10=92∘3(34)-10 = 92^\circ. Check: 88+92=18088+92 = 180. ✓

Tip

Watch the word: “congruent” ⇒\Rightarrow set the expressions equal. “Supplementary” ⇒\Rightarrow set their sum equal to 180180.

Proving Lines Are Parallel (the Converses)

The rules above run backwards too. If you can find one of the special angle relationships, you can conclude the lines are parallel.

Two parallel lines cut by a transversal.

Concept
The converse statements

Two lines cut by a transversal are parallel if any one of these is true:

  • [leftmargin=6mm,itemsep=1pt]
  • a pair of corresponding angles is congruent, or
  • a pair of alternate interior angles is congruent, or
  • a pair of alternate exterior angles is congruent, or
  • a pair of same-side interior angles is supplementary.
Example
Are these lines parallel?

A transversal crosses two lines. A pair of alternate interior angles measure 63∘63^\circ and 63∘63^\circ. Because these alternate interior angles are congruent, the converse rule tells us the two lines are parallel. If instead they had measured 63∘63^\circ and 70∘70^\circ, they would not be congruent, so the lines would not be parallel.

Tip

Careful: Same-side interior angles do the opposite --- the lines are parallel when those two angles add to 180∘180^\circ, not when they are equal.

Perpendicular Lines and Right Angles

Concept
Perpendicular means right angles
55°

When two lines are perpendicular (ℓ⊥m\ell \perp m), they cross to make four 90∘90^\circ angles. A small square drawn in the corner is the symbol for a right angle. If one of the four angles is 90∘90^\circ, all four are.

An angle measures a turn.

Example
Splitting a right angle

A right angle is split by a ray into two smaller angles of x∘x^\circ and 52∘52^\circ. Since the whole angle is 90∘90^\circ:

x+52=90⇒x=38∘.x + 52 = 90 \quad\Rightarrow\quad x = 38^\circ.
Tip

Remember: Perpendicular is the same idea whether you are looking at drawn lines (right angles) or graphed lines (slopes that are negative reciprocals --- next section).

Slopes of Parallel and Perpendicular Lines

On the coordinate plane, slope measures steepness: slope=riserun=change in ychange in x\text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{\text{change in }y}{\text{change in }x}.

Two parallel lines cut by a transversal.

Concept
The slope rules

Parallel lines have equal slopes. If line 1 has slope mm, then any line parallel to it also has slope mm. Perpendicular lines have slopes that are negative reciprocals. Flip the fraction and change the sign: if one slope is mm, the perpendicular slope is −1m-\dfrac{1}{m}. A quick test: two slopes are perpendicular exactly when their product is −1-1.

Example
Read off the slopes

Line AA has slope 34\dfrac{3}{4}.

  • [leftmargin=6mm,itemsep=1pt]
  • A line parallel to AA has slope 34\dfrac{3}{4} (same).
  • A line perpendicular to AA has slope −43-\dfrac{4}{3} (flip 34\tfrac34 to 43\tfrac43, change the sign). Check: 34⋅(−43)=−1\dfrac34 \cdot \left(-\dfrac43\right) = -1. ✓
Example
Parallel, perpendicular, or neither?

Line AA has slope −2-2 and line BB has slope 12\dfrac{1}{2}. Their product is −2⋅12=−1-2 \cdot \dfrac12 = -1, so the lines are perpendicular.

Tip

Special cases: A horizontal line has slope 00; a vertical line has an undefined slope. A horizontal line and a vertical line are perpendicular to each other.

Writing Equations of Parallel and Perpendicular Lines

We use slope-intercept form y=mx+by = mx + b and point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1).

Two parallel lines cut by a transversal.

Concept
The three-step recipe
  • [leftmargin=6mm,itemsep=1pt]
  • Find the slope of the given line (read mm from y=mx+by=mx+b).
  • Choose the new slope: same mm for parallel, negative reciprocal for perpendicular.
  • Plug that slope and the given point into point-slope form, then simplify to y=mx+by=mx+b.
Example
A parallel line through a point

Write the equation of the line through (1,4)(1,4) that is parallel to y=2x+1y = 2x + 1. The given slope is 22; parallel means the new slope is also 22. Point-slope:

y−4=2(x−1)  ⇒  y−4=2x−2  ⇒  y=2x+2.y - 4 = 2(x - 1) \;\Rightarrow\; y - 4 = 2x - 2 \;\Rightarrow\; y = 2x + 2.

Check the point: 2(1)+2=42(1)+2 = 4. ✓

Example
A perpendicular line through a point

Write the equation of the line through (4,3)(4,3) that is perpendicular to y=2x+1y = 2x + 1. The given slope is 22; the perpendicular slope is −12-\dfrac12. Point-slope:

y−3=−12(x−4)  ⇒  y−3=−12x+2  ⇒  y=−12x+5.y - 3 = -\tfrac12(x - 4) \;\Rightarrow\; y - 3 = -\tfrac12 x + 2 \;\Rightarrow\; y = -\tfrac12 x + 5.

Check the point: −12(4)+5=3-\tfrac12(4)+5 = 3. ✓

Tip

Final tip: Always test your equation by plugging the given point back in. If it makes the equation true, your line really does pass through that point --- and the slope guarantees it is parallel or perpendicular as required.

Going Deeper: Advanced Ideas

Up to now we have used the parallel-line rules. Here we see why they are true, chase angles through several transversals at once, and push the slope ideas into real coordinate computations: distances, feet of perpendiculars, and equations of lines.

An angle measures a turn.

Concept
Why the transversal theorems are true

The whole system rests on one assumed fact, the Corresponding Angles Postulate: if two parallel lines are cut by a transversal, corresponding angles are congruent. Everything else is a short deduction from it, using two facts you already know:

  • [leftmargin=6mm,itemsep=1pt]
  • Vertical angles (across an X) are congruent.
  • A linear pair (two angles on a straight line) is supplementary.

Chain them together: a corresponding angle is congruent to the one directly across from it (vertical), which is the alternate interior angle --- so alternate interior angles are congruent. Swap the vertical step for a linear pair and you get same-side interior angles are supplementary. No new assumptions are needed.

Example
Worked proof: alternate interior angles are congruent

Using the eight-angle numbering from earlier, suppose m∥nm \parallel n. Prove ∠3≅∠6\angle 3 \cong \angle 6.

The angle you start with (∠3\angle 3) hops to its corresponding partner (∠7\angle 7), then across the X to its vertical partner (∠6\angle 6). Two congruences in a row give the result. The same two-step hop, run backwards, proves the converse used to show lines are parallel.

Concept
The auxiliary parallel line trick (zigzag chases)

When a path bends between two parallel lines, there is no single transversal to use. The fix: draw a new line through the bend point, parallel to the two given lines. This splits the bend angle into two pieces, and each piece is now an alternate-interior angle with one of the original lines. Add (or subtract) the pieces to finish. This “build your own parallel line” move turns an impossible-looking figure into two ordinary transversal problems.

Example
Worked zigzag angle chase

Lines rr and ss are parallel. A path goes from AA on rr down to a corner PP, then to BB on ss. The path makes a 35∘35^\circ angle with rr at AA and a 25∘25^\circ angle with ss at BB. Find ∠APB\angle APB.

Draw the dashed line through PP parallel to rr and ss. It splits ∠APB\angle APB into an upper piece and a lower piece.

  • [leftmargin=6mm,itemsep=1pt]
  • Upper piece =35∘=35^\circ: alternate interior angles with AA's angle across transversal APAP.
  • Lower piece =25∘=25^\circ: alternate interior angles with BB's angle across transversal BPBP.

So ∠APB=35∘+25∘=60∘\angle APB = 35^\circ + 25^\circ = \mathbf{60^\circ}. (Whenever the bend points away from the region between the parallels you add the pieces; if the path stays on one side you subtract them instead.)

Concept
Distance between two parallel lines

Parallel lines never meet, so they stay a fixed distance apart --- but that distance is measured perpendicular to the lines, not vertically. For two lines written as

y=mx+b1andy=mx+b2(same slope m),y = mx + b_1 \qquad\text{and}\qquad y = mx + b_2 \quad(\text{same slope } m),

the perpendicular distance between them is

d=∣b2−b1∣1+m2.d = \frac{\lvert b_2 - b_1 \rvert}{\sqrt{1 + m^2}}.

The numerator is the vertical gap between the intercepts; dividing by 1+m2\sqrt{1+m^2} tilts that vertical gap into the true perpendicular gap. (If both lines are given in the form ax+by+c=0ax+by+c=0 with matching a,ba,b, the distance is ∣c2−c1∣/a2+b2\lvert c_2 - c_1\rvert / \sqrt{a^2+b^2}.)

Example
Worked distance between parallels

Find the distance between y=2x+1y = 2x + 1 and y=2x+6y = 2x + 6. Same slope m=2m = 2, so they are parallel. Here b1=1b_1 = 1 and b2=6b_2 = 6:

d=∣6−1∣1+22=55=5≈2.24.d = \frac{\lvert 6 - 1 \rvert}{\sqrt{1 + 2^2}} = \frac{5}{\sqrt{5}} = \sqrt{5} \approx 2.24.

Notice this is less than the vertical gap of 55 --- the perpendicular route between two slanted lines is always the shortest one.

Concept
The foot of a perpendicular

The foot of the perpendicular from a point PP to a line ℓ\ell is the single point FF on ℓ\ell where the perpendicular dropped from PP lands. It is the closest point on ℓ\ell to PP, and the distance PFPF is the true distance from the point to the line. To find it:

  • [leftmargin=6mm,itemsep=1pt]
  • Take the slope of ℓ\ell and use its negative reciprocal as the slope of the perpendicular through PP.
  • Write that perpendicular line in point-slope form.
  • Solve the two equations together --- the solution is the foot FF.
Example
Worked foot of a perpendicular

Find the foot of the perpendicular from P=(4,1)P = (4,1) to the line ℓ: y=x+1\ell:\ y = x + 1, and the distance from PP to ℓ\ell. Line ℓ\ell has slope 11, so the perpendicular has slope −1-1. Through P=(4,1)P=(4,1):

y−1=−1(x−4)  ⇒  y=−x+5.y - 1 = -1(x - 4) \;\Rightarrow\; y = -x + 5.

Set the two lines equal to find where they cross:

x+1=−x+52x=4x=2,y=2+1=3.\begin{aligned} x + 1 &= -x + 5\\ 2x &= 4\\ x &= 2, \qquad y = 2 + 1 = 3. \end{aligned}

The foot is F=(2,3)F = (2,3). The distance is

PF=(4−2)2+(1−3)2=4+4=8=22≈2.83.PF = \sqrt{(4-2)^2 + (1-3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83.

No other point on ℓ\ell is closer to PP than FF.

Tip

Why the slope criteria are consistent: a line with slope ba\tfrac{b}{a} points in direction (a,b)(a,b). Rotating that direction a quarter turn (90∘90^\circ) sends (a,b)↦(−b,a)(a,b) \mapsto (-b,a), whose slope is a−b=−ab\dfrac{a}{-b} = -\dfrac{a}{b} --- exactly the negative reciprocal, and their product is −1-1. So the algebra (negative reciprocals) and the geometry (a right angle) are the same statement. Likewise, two lines have equal slopes precisely when they point the same direction, which is precisely when they never cross --- the definition of parallel.

Formulas, Proofs & Tips

Tip
Slope of a line
m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}

What it means. Rise over run: how much yy changes for each 11 that xx increases.

Example. Through (1,2)(1,2) and (4,8)(4,8): m=8−24−1=2m=\dfrac{8-2}{4-1}=2.

Why it works. Between two points on a line the vertical change is y2−y1y_2-y_1 and the horizontal change is x2−x1x_2-x_1. Similar triangles guarantee this ratio is the same wherever you measure it, so it is a property of the line itself.

Tip. Keep the points in the same order top and bottom. Reversing both gives the same slope; reversing only one flips the sign.

Tip
Angle pair relationships
vertical angles are equal;linear pair sums to 180∘;complements sum to 90∘\text{vertical angles are equal};\quad \text{linear pair sums to }180^\circ;\quad \text{complements sum to }90^\circ

What it means. Angles formed by crossing lines come in predictable pairs.

Example. If two angles are supplementary and one is 110∘110^\circ, the other is 70∘70^\circ.

Why it works. Two angles on a straight line total 180∘180^\circ. If ∠1\angle 1 and ∠2\angle 2 form a line, and ∠2\angle 2 and ∠3\angle 3 also form a line, then both equal 180∘−∠2180^\circ-\angle 2 — so the vertical angles ∠1\angle 1 and ∠3\angle 3 are equal.

Tip. With parallel lines cut by a transversal, corresponding and alternate angles are equal, while same-side interior angles are supplementary.

Tip
Parallel and perpendicular slopes
m1=m2 (parallel),m1m2=−1 (perpendicular)m_1=m_2 \ (\text{parallel}), \qquad m_1m_2=-1 \ (\text{perpendicular})

What it means. Parallel lines share a slope; perpendicular slopes are opposite reciprocals.

Example. A line perpendicular to slope 22 has slope −12-\tfrac12 (their product is −1-1).

Why it works. Rotating a direction by 90∘90^\circ turns a run of aa and rise of bb into a run of −b-b and rise of aa, changing slope ba\tfrac{b}{a} into −ab-\tfrac{a}{b} — whose product with the original is −1-1.

Tip. A horizontal line (m=0m=0) is perpendicular to a vertical line, whose slope is undefined — the product rule does not apply to that pair.