To determine if two lines are parallel, we need to compare their slopes. The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \((y_2 - y_1) / (x_2 - x_1)\). If the slopes of the two lines are equal, then the lines are parallel.
For Exercise 19, calculate the slope of the line through points \(A(1,-2)\) and \(B(-3,-10)\), and the slope of the line through points \(C(1,5)\) and \(D(-1,1)\). Compare these slopes to determine if the lines are parallel.
For Exercise 20, calculate the slope of the line through points \(A(2,3)\) and \(B(2,-2)\), and the slope of the line through points \(C(-2,4)\) and \(D(-2,6)\). Compare these slopes to determine if the lines are parallel.
The slope \(m_1\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\]
For points \(A(1, -2)\) and \(B(-3, -10)\):
\[
m_1 = \frac{-10 - (-2)}{-3 - 1} = \frac{-8}{-4} = 2
\]
For points \(C(1, 5)\) and \(D(-1, 1)\):
\[
m_2 = \frac{1 - 5}{-1 - 1} = \frac{-4}{-2} = 2
\]
Two lines are parallel if their slopes are equal. Since \(m_1 = 2\) and \(m_2 = 2\), the lines through points \(A\) and \(B\), and points \(C\) and \(D\) are parallel.
For points \(A(2, 3)\) and \(B(2, -2)\), the line is vertical, so the slope is undefined. We represent this as:
\[
m_3 = \infty
\]
For points \(C(-2, 4)\) and \(D(-2, 6)\), the line is also vertical, so the slope is:
\[
m_4 = \infty
\]
Since both lines are vertical, they are parallel.
\(\boxed{\text{The lines in both exercises are parallel.}}\)
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