Question

I. Two non-vertical lines $l_1$ and $l_2$ are parallel, if and only if their slopes are equal.
II. Two non-vertical lines $l_1$ and $l_2$ are perpendicular to each other, if and only if their slopes are negative reciprocals of each other.

• A. I and II are true
• B. I and II are false
• C. I is true and II is false
• D. II is true and I is false

Answer 1

Answer: (A)

In a coordinate plane, suppose that non-vertical lines $l_1$ and $l_2$ have slopes $m_1$ and $m_2$, respectively. Let their inclination be $\alpha$ and $\beta$, respectively.
If the line $l_1$ is parallel to $l_2$.
Then, their inclinations are equal i.e.,

$\alpha=\beta$, and hence $\tan \alpha=\tan \beta$
Therefore, $m_1=m_2$, i.e., their slopes are equal.
Conversely, if the slope of two lines $l_1$ and $l_2$ is same i.e.,
$ m_1=m_2 $
Then, $ \tan \alpha=\tan \beta $
By the property of tangent function (between $ 0^{\circ} $ and $ 180^{\circ}),$
$ \alpha=\beta$
Therefore, the lines are parallel.
Hence, two non-vertical lines $l_1$ and $l_2$ are parallel, if and only if their slopes are equal.
If the lines $l_1$ and $l_2$ are perpendicular .
Here, $ m_1=\tan \alpha \left(\because m_1\right.$ is the slope of $\left.l_1\right)$

$\beta$ is the exterior angle of triangle which is equal to the sum of interior opposite angles of triangle i.e., $\alpha$ and $90^{\circ}$.
Therefore, $\tan \beta=\tan \left(\alpha+90^{\circ}\right)$
$ =-\cot \alpha $
$ =-\frac{1}{\tan \alpha}$
i.e., $ m_2=-\frac{1}{m_1}$ or $m_1 m_2=-1$
Conversely, if $m_1 m_2=-1$ i.e., $\tan \alpha \tan \beta=-1$.
Then, $\tan \alpha=-\cot \beta=\tan \left(\beta+90^{\circ}\right)$ or $\tan \left(\beta-90^{\circ}\right)$.
Therefore, $\alpha$ and $\beta$ differ by $90^{\circ}$.
Thus, lines $l_1$ and $l_2$ are perpendicular to each other.
Hence, two non-vertical lines are perpendicular to each other, if and only if their slopes are negative reciprocals of each other.
i.e., $ m_2=-\frac{1}{m_1}$ or $m_1 m_2=-1$