Question

talk the talk
topic 1
parallels the lesson
previously, you analyzed a worked example that demonstrates \when two lines are perpendicular, their slopes are opposite reciprocals of each other.\
1 using similar reasoning, write an explanation that justifies \when two lines are parallel, their slopes are equal.\ include a sketch.
assumption: p||q
let (m_1 =) slope of line p.
let (m_2 =) slope of line q.
conclusion: (m_1 = m_2)

Explanation:

Step1: Recall slope - angle relationship

The slope \(m\) of a non - vertical line is given by \(m = \tan\theta\), where \(\theta\) is the angle the line makes with the positive \(x\) - axis (\(0\leq\theta<\pi\)).

Step2: Consider parallel lines

If two lines \(p\) and \(q\) are parallel (\(p\parallel q\)), then the angles \(\theta_1\) and \(\theta_2\) that they make with the positive \(x\) - axis are equal, i.e., \(\theta_1=\theta_2\).

Step3: Calculate slopes

Since \(m_1 = \tan\theta_1\) and \(m_2=\tan\theta_2\), and \(\theta_1 = \theta_2\), we have \(m_1=m_2\).

For the sketch:

1. Draw two parallel lines \(p\) and \(q\) on the given coordinate grid. For example, draw a line \(p\) passing through points \((0,0)\) and \((1,1)\) with slope \(m_1=\frac{1 - 0}{1 - 0}=1\). Then draw a line \(q\) passing through points \((0,2)\) and \((1,3)\) with slope \(m_2=\frac{3 - 2}{1 - 0}=1\). The two lines will be parallel and have the same steepness.

Answer:

The slopes of two parallel lines are equal because the angles that the parallel lines make with the positive \(x\) - axis are equal, and the slope of a line is the tangent of the angle it makes with the positive \(x\) - axis. A sketch can be made by drawing two lines with the same steepness on the coordinate grid.