Question

11 multiple choice 1 point
in the figure, line m is parallel to line n. which angle must have the same measure as ∠x?
∠3
∠4
∠5
∠2
12 multiple choice 1 point
which of the following systems represent a pair of lines that are perpendicular to each other?
2x + 3y = 6
-3x + 2y = 6
2x + 3y = 6
3x + 2y = 6
3x + 2y = 6
-3x + 2y = 6
3x + 2y = 6
6x + 4y = 12

Explanation:

Step1: Analyze the relationship between angles for question 11

When two parallel lines \(m\) and \(n\) are cut by a transversal, \(\angle x\) and \(\angle2\) are alternate - interior angles. By the Alternate - Interior Angles Theorem, if two parallel lines are cut by a transversal, then alternate - interior angles are congruent.

Step2: Analyze the slopes of lines for question 12

The equation of a line is \(y = mx + b\), where \(m\) is the slope. For a line \(Ax+By = C\), we can rewrite it as \(y=-\frac{A}{B}x+\frac{C}{B}\).

• For the line \(2x + 3y=6\), rewrite it as \(y=-\frac{2}{3}x + 2\), slope \(m_1=-\frac{2}{3}\).
• For the line \(-3x + 2y=6\), rewrite it as \(y=\frac{3}{2}x+3\), slope \(m_2=\frac{3}{2}\).
• For the line \(3x + 2y=6\), rewrite it as \(y=-\frac{3}{2}x + 3\), slope \(m_3=-\frac{3}{2}\).
• For the line \(6x + 4y=12\), rewrite it as \(y=-\frac{6}{4}x+3=-\frac{3}{2}x + 3\), slope \(m_4=-\frac{3}{2}\).

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\). We check: \(-\frac{2}{3}\times\frac{3}{2}=-1\)

Answer:

1. \(\angle2\)
2. \(2x + 3y = 6\) and \(-3x + 2y = 6\)