Question

nes mn, op, qr, and st all intersect at a common point, z.
elect each statement that provides enough information to conclude that \overleftrightarrow{op} is p\overleftrightarrow{qr}.
\checkmark \\ m\angle ozt - m\angle qzt = 90^\circ
\checkmark \\ \angle pzr is a right angle.
\checkmark \\ m\angle mzs = 90^\circ and m\angle ozs = 30^\circ.
\checkmark \\ m\angle mzo + m\angle ozs = 90^\circ

Explanation:

To determine which statements show \(\overleftrightarrow{OP}\perp\overleftrightarrow{QR}\), we use the definition of perpendicular lines (they form a \(90^\circ\) angle).

1. Analyze \(m\angle OZT - m\angle QZT = 90^\circ\)

Let \(\angle QZT = x\). Then \(m\angle OZT = x + 90^\circ\). The angle between \(\overleftrightarrow{OP}\) and \(\overleftrightarrow{QR}\) is \(m\angle OZT - m\angle QZT = 90^\circ\), so they are perpendicular.

2. Analyze \(\angle PZR\) is a right angle

If \(\angle PZR = 90^\circ\), by definition of perpendicular lines, \(\overleftrightarrow{OP}\perp\overleftrightarrow{QR}\) (since \(\angle PZR\) is the angle between them).

3. Analyze \(m\angle MZS = 90^\circ\) and \(m\angle OZS = 30^\circ\)

Since \(m\angle MZS = 90^\circ\) (a right angle) and \(m\angle OZS = 30^\circ\), we find \(m\angle MZO = 90^\circ - 30^\circ = 60^\circ\). Using vertical angles or linear pairs (depending on the diagram), we can show the angle between \(\overleftrightarrow{OP}\) and \(\overleftrightarrow{QR}\) is \(90^\circ\), so they are perpendicular.

4. Analyze \(m\angle MZO + m\angle OZS = 90^\circ\)

This only tells us \(m\angle MZS = 90^\circ\), but it does not directly relate to the angle between \(\overleftrightarrow{OP}\) and \(\overleftrightarrow{QR}\). Thus, it does not prove perpendicularity.

The statements that provide enough information are:

• \(m\angle OZT - m\angle QZT = 90^\circ\)
• \(\angle PZR\) is a right angle.
• \(m\angle MZS = 90^\circ\) and \(m\angle OZS = 30^\circ\).

Answer:

• \(m\angle OZT - m\angle QZT = 90^\circ\)
• \(\angle PZR\) is a right angle.
• \(m\angle MZS = 90^\circ\) and \(m\angle OZS = 30^\circ\)