Angle Between Lines, Parallel & Perpendicular

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 $-a_i/b_i$.

• Same formula, other forms. General form $lx+my+n=0$, $l'x+m'y+n'=0$: $\tan\theta=\left|\dfrac{lm'-l'm}{ll'+mm'}\right|$ — the option lists print it this way. Normal form $x\cos\alpha+y\sin\alpha=p$, $x\cos\beta+y\sin\beta=q$: the angle between the lines is simply $|\alpha-\beta|$ (the angle between their normals).
• Angle subtended at the origin by two POINTS $(m,n)$ and $(r,s)$ — a dot-product question, not a slope one: $\cos\alpha=\dfrac{mr+ns}{\sqrt{(m^2+n^2)(r^2+s^2)}}$, and $\sin\alpha=\dfrac{|ms-nr|}{\sqrt{(m^2+n^2)(r^2+s^2)}}$. The planted wrong option swaps the two numerators — $mr+ns$ belongs to the cosine, $ms-nr$ to the sine.

• Parallel: $m_1=m_2$. In coefficients, for $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, that is $a_1b_2=a_2b_1$, i.e. $\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}$.
• That test also passes for the same line written twice. It says the two lines have the same direction; it does not say they are different lines. They are coincident when the constants agree as well, $\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}=\tfrac{c_1}{c_2}$, and strictly parallel when the first two ratios agree and the third does not. A question that says "parallel" almost always wants the strict kind — so check the constant before you answer, or you will pick a value that makes the two lines identical.
• Perpendicular: $m_1 m_2=-1$, i.e. $a_1a_2+b_1b_2=0$.