Question

identify all pairs of parallel and perpendicular lines in the image below.
(image of intersecting lines with labels a, b, c, d, e, f, g, h, i, j)
multiple choice options (four options with line relationships: perpendicular and parallel pairs)

Explanation:

Step1: Analyze Perpendicular Lines

Perpendicular lines form a right angle (90°). From the diagram, lines with right - angle symbols: \(CD\perp EF\) (right angle symbol between them), \(CD\perp IJ\) (right angle symbol), \(AB\perp GH\) (right angle symbol). Also, since \(EF\parallel IJ\) (same slope, vertical lines), and \(CD\) is perpendicular to \(EF\), it's perpendicular to \(IJ\) too. \(AB\) is perpendicular to \(GH\), and since \(EF\parallel IJ\parallel GH\)? Wait, no, vertical lines: \(EF\), \(GH\), \(IJ\) are vertical (same direction), horizontal lines: \(AB\), \(CD\) are horizontal (same direction). So horizontal lines are parallel (\(AB\parallel CD\)), vertical lines are parallel (\(EF\parallel GH\parallel IJ\)). Perpendicular: horizontal \(\perp\) vertical. So \(CD\perp EF\), \(CD\perp IJ\), \(AB\perp GH\), \(AB\perp EF\) (since \(AB\) is horizontal, \(EF\) vertical), \(AB\perp IJ\) (same reason). Wait, but let's check the options. The last option has \(CD\perp EF\), \(CD\perp IJ\), \(AB\perp GH\), \(EF\parallel IJ\), \(EF\parallel GH\) (since all vertical). Wait, the options: Let's parse the last option (the fourth circle, first option? Wait, the options are:

1. \(CD\perp EF\), \(CD\perp IJ\), \(EF\parallel IJ\)
2. \(CD\perp EF\), \(EF\perp AB\), \(AB\perp GH\), \(EF\parallel IJ\) (wait, no, the second option as per text: \(CD\perp EF\), \(EF\perp AB\)? No, the second option text: \(CD\perp EF\), \(EF\perp AB\)? Wait, the user's text for options:

First option (first circle): \(CD\perp EF\), \(CD\perp IJ\), \(EF\parallel IJ\)

Second option (second circle): \(CD\perp EF\), \(CD\perp IJ\), \(AB\perp GH\), \(EF\parallel IJ\)

Third option (third circle): \(CD\perp EF\), \(EF\perp AB\), \(AB\perp GH\), \(EF\parallel IJ\) (no, the third option text: \(CD\perp EF\), \(EF\perp AB\)? Wait, the original text for options:

First option: \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{IJ}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\)

Second option: \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{IJ}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{GH}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\)

Third option: \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{EF}\perp\overleftrightarrow{AB}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{GH}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\)

Fourth option (last circle, fourth option? Wait, the last option (the one with more relations) is: \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{IJ}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{GH}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{GH}\) (the fourth option, the last circle's text: \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{IJ}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{GH}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{GH}\))

Wait, horizontal lines: \(AB\) and \(CD\) are parallel (\(AB\parallel CD\)). Vertical lines: \(EF\), \(GH\), \(IJ\) are parallel (\(EF\parallel GH\parallel IJ\)). Perpendicular: horizontal \(\perp\) vertical, so \(AB\perp EF\), \(AB\perp GH\), \(AB\perp IJ\), \(CD\perp EF\), \(CD\perp GH\), \(CD\perp IJ\). Parallel: \(AB\parallel CD\), \(EF\parallel GH\parallel IJ\).

Now check the options:

The fourth option (the one with the most relations) has \(CD\perp EF\), \(CD\perp IJ\), \(AB\perp GH\), \(EF\parallel IJ\), \(EF\parallel GH\). Let's verify:

- \(CD\perp EF\): horizontal \(\perp\) vertical: yes.
- \(CD\perp IJ\): horizontal \(\perp\) vertical: yes.
- \(AB\perp GH\): horizontal \(\perp\) vertical: yes.
- \(EF\parallel IJ\): both vertical: yes.
- \(EF\parallel GH\): both vertical: yes.

So this option includes all correct perpendicular and parallel pairs.

Step2: Confirm the Option

The last option (the fourth circle, the option with \(CD\perp EF\), \(CD\perp IJ\), \(AB\perp GH\), \(EF\parallel IJ\), \(EF\parallel GH\)) is correct.

Answer:

The option with the fourth circle (the last option) which has \(\overleftrightarrow{CD}\perp\overleftrightarrow{EF}\), \(\overleftrightarrow{CD}\perp\overleftrightarrow{IJ}\), \(\overleftrightarrow{AB}\perp\overleftrightarrow{GH}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{IJ}\), \(\overleftrightarrow{EF}\parallel\overleftrightarrow{GH}\) (the fourth option in the list of circles, the one with the most relations).