NDA Maths · Lines

Angle Between Lines, Parallel & Perpendicular

The angle between two lines from their slopes, and the slope conditions for lines to be parallel or perpendicular.

Why this matters

The angle formula and the parallel/perpendicular tests are short, high-frequency tools — used directly and inside triangle and quadrilateral problems. The only trap is the sign in the tangent formula (acute vs obtuse).

Concept 1 of 2

Angle between two lines

Intuition

The angle between two lines depends only on their slopes. The tangent formula gives the acute angle when you take the absolute value; drop the bars (or take the supplement) for the obtuse one.

Definition

For slopes $m_1,m_2$ : $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right|$ gives the acute angle; the obtuse angle is its supplement. If $1+m_1m_2=0$ the lines are perpendicular ( $\theta=90°$ ). For lines given as $a_1x+b_1y+c_1=0$ , use slopes $-a_i/b_i$ .

Angle between two lines

\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right|

Worked example

Find the acute angle between lines of slopes

1

and

\tfrac13

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1LinesMODERATE

What is the obtuse angle between the lines whose slopes are

2-\sqrt{3}

and

2+\sqrt{3}

[Q59 · Apr · 2020]

The difference of slopes is on top: $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$

The angle formula puts the difference

m_1-m_2

in the numerator and

1+m_1m_2

in the denominator — students often invert it to

\dfrac{1+m_1m_2}{m_1-m_2}

. Also watch the denominator's plus sign (

1+m_1m_2

, not

1-m_1m_2

); when

1+m_1m_2=0

the tangent blows up, correctly signalling

\theta=90^\circ

Drill 11 more on angle between two lines

Concept 2 of 2

Parallel and perpendicular conditions

Intuition

Two lines are parallel when their slopes match, and perpendicular when the slopes multiply to

-1

. In coefficient form these become clean conditions on

a,b

Definition

Parallel: $m_1=m_2$ ; for $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ , parallel iff $a_1b_2=a_2b_1$ (i.e. $\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}$ ). Perpendicular: $m_1 m_2=-1$ , i.e. $a_1a_2+b_1b_2=0$ .

Parallel and perpendicular conditions

\text{Parallel: } m_1=m_2\qquad \text{Perpendicular: } m_1 m_2=-1\qquad a_1a_2+b_1b_2=0

Worked example

Are

2x+3y=5

and

3x-2y=7

perpendicular?

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2LinesMODERATE

Under which one of the following conditions are the lines

ax+by+c=0

and

bx+ay+c=0

parallel (

a\neq0, b\neq0

[Q88 · Apr · 2022]

Perpendicular slope is the negative reciprocal: $m_2=-\dfrac{1}{m_1}$ , not $\dfrac{1}{m_1}$

Parallel ⇒ equal slopes (

m_1=m_2

); perpendicular ⇒ the product is

-1

(

m_1m_2=-1

), so the second slope is the negative reciprocal

-1/m_1

. The two classic slips: forgetting the minus (using

1/m_1

, the plain reciprocal), and swapping the two rules — "perpendicular means equal slopes" is wrong. If

m_1=\tfrac23

, a perpendicular line has slope

-\tfrac32

, not

\tfrac32

Drill 3 more on parallel and perpendicular conditions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Angle between two lines
Angle between two lines
$\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1 m_2}\right|$
Parallel and perpendicular conditions
Parallel and perpendicular conditions
$\text{Parallel: } m_1=m_2\qquad \text{Perpendicular: } m_1 m_2=-1\qquad a_1a_2+b_1b_2=0$

Watch out for (2)

The difference of slopes is on top: $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$
→ Angle between two lines
Perpendicular slope is the negative reciprocal: $m_2=-\dfrac{1}{m_1}$ , not $\dfrac{1}{m_1}$
→ Parallel and perpendicular conditions

Drill every past-year question on this subtopic

16 questions from the bank — paginated, with cart and Word-export support.

Drill the 16 questions

Drill every past-year question on this subtopic

Related notes