Slopes of parallel lines — lesson. Mathematics State Board, Class 10.

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Let

$l_1$ and

$l_2$ be two non-vertical lines.

The slope of line

$l_1$ is

$m_1$ , and line

$l_2$ is

$m_2$ .

Let the inclination of

$l_1$ be

$\theta_1$ and

$l_2$ be

$\theta_2$ .

Assume

$l_1$ and

$l_2$ are parallel lines.

If two lines are parallel, then their corresponding angles are equal.

$\Rightarrow \theta_1 = \theta_2$

$\Rightarrow \tan \theta_1 = \tan \theta_2$

$\Rightarrow m_1 = m_2$

The slopes are equal.

Conversely:

Assume slopes of two lines

$l_1$ and

$l_2$ , are equal.

$\Rightarrow m_1 = m_2$

$\Rightarrow \tan \theta_1 = \tan \theta_2$

$\Rightarrow \theta_1 = \theta_2$

$\Rightarrow$ Corresponding angles are equal.

$\Rightarrow$

$l_1$ and

$l_2$ are parallel.

Thus, the non-vertical lines are parallel if and only if their slopes are equal.

If two lines are parallel, then their slopes are equal. That is,

$m_1 = m_2$ .

Example:

If a line

$p$ passing through the points

$(1, 8)$ and

$(2, 13)$ and a line

$q$ passing through the points

$(0, -1)$ and

$(1, 4)$ are parallel?

Solution:

Let the points passing through the line

$p$ be

$A$

$=$

$(1, 8)$ and

$B = (2, 13)$ .

And, the points passing through the line

$q$ be

$C = (0, -1)$ and

$D = (1, 4)$ .

Two lines are parallel if their slopes are equal.

Let us find the slopes of

$p$ and

$q$ .

Slope

$=$

$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Slope of

$p$

$=$

$\frac{13 - 8}{2 - 1} = \frac{5}{1} = 5$

Slope of

$q$

$=$

$\frac{4 + 1}{1 - 0} = \frac{5}{1} = 5$

Hence, the slope of

$p$

$=$ slope of

$q$ .

Therefore, the lines

$p$ and

$q$ are parallel.

Important!

The quadrilateral is a parallelogram if the slopes of both pairs of opposite sides are equal.

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4. Slopes of parallel lines

Theory: