Question

given: vertical line a and horizontal line b
prove: a ⊥ b

1. the x - axis and y - axis are

2.
3.

1. a is
2. a ⊥ b

word bank:
perpendicular transversal theorem b // x - axis
b // y - axis perpendicular
perpendicular to the y - axis
b is perpendicular to the x - axis
parallel transversal theorem a // y - axis
perpendicular to the x - axis a // x - axis parallel

Explanation:

Step1: Recall coordinate - axis property

The x - axis and y - axis are perpendicular.

Step2: State line - axis parallel relationships

\(a\parallel y - axis\) (vertical line \(a\) is parallel to the y - axis) and \(b\parallel x - axis\) (horizontal line \(b\) is parallel to the x - axis).

Step3: Use Perpendicular Transversal Theorem

If a line is parallel to one of two perpendicular lines, then it is perpendicular to the other. Since \(a\parallel y - axis\) and \(b\parallel x - axis\), and \(x - axis\perp y - axis\), by the Perpendicular Transversal Theorem.

Step4: State line - axis perpendicular relationships

\(a\) is perpendicular to the x - axis (because \(a\parallel y - axis\) and \(y - axis\perp x - axis\)), and \(b\) is perpendicular to the x - axis (because \(b\parallel x - axis\) is wrong, actually \(b\parallel x - axis\) and \(x - axis\perp y - axis\), so \(b\) is perpendicular to \(y - axis\) in another way of looking, but more accurately, using the parallel and perpendicular relationships).

Answer:

1. Perpendicular
2. \(a\parallel y - axis\), \(b\parallel x - axis\)
3. Perpendicular Transversal Theorem
4. perpendicular to the x - axis