Question

determine if each pair of lines is parallel. see example 2
graph with lines j, k, m, n, p, q, and coordinate axes (x, y) with grid

1. j and k
2. m and n
3. p and q

Explanation:

20. Determine if lines \( j \) and \( k \) are parallel

Step1: Analyze the slope of vertical lines

Vertical lines have an undefined slope. Both lines \( j \) and \( k \) are vertical (they run straight up and down, parallel to the \( y \)-axis).

Step2: Compare slopes

Since both are vertical lines, their slopes are both undefined, and vertical lines with undefined slopes are parallel.

Step1: Find two points on line \( m \)

For line \( m \), let's take points \((0, 6)\) and \((2, 10)\) (by moving along the grid). The slope \( m_m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{10 - 6}{2 - 0}=\frac{4}{2} = 2\).

Step2: Find two points on line \( n \)

For line \( n \), take points \((4, 4)\) and \((6, 8)\). The slope \( m_n=\frac{8 - 4}{6 - 4}=\frac{4}{2}=2\).

Step3: Compare slopes

Since \( m_m = m_n=2\), lines with equal slopes are parallel.

Step1: Find two points on line \( p \)

For line \( p \), take points \((0, 6)\) and \((6, 4)\). The slope \( m_p=\frac{4 - 6}{6 - 0}=\frac{-2}{6}=-\frac{1}{3}\).

Step2: Find two points on line \( q \)

For line \( q \), take points \((0, -2)\) and \((6, -4)\). The slope \( m_q=\frac{-4 - (-2)}{6 - 0}=\frac{-2}{6}=-\frac{1}{3}\).

Step3: Compare slopes

Since \( m_p = m_q = -\frac{1}{3}\), lines with equal slopes are parallel.

Answer:

Lines \( j \) and \( k \) are parallel.

21. Determine if lines \( m \) and \( n \) are parallel