Question

myra took a picture of the sky one afternoon when two jet airplanes appeared to draw a pair of parallel lines with their vapor trails. the vapor trails from two other jets flying from another direction crossed over the parallel trails. she printed her picture and labeled the angles and lines.
assume lines c and d are parallel and ∠2 measures 98°. which statements are true? select three options.
□ m∠3 = m∠6 = 98°
□ m∠3 = m∠14 = 98°
□ m∠4 = m∠8 = 82°
□ m∠4 = m∠12 = 82°
□ m∠5 = m∠8 = 82°

Explanation:

Step1: Analyze ∠2 and related angles

Given \( m\angle2 = 98^\circ \). Vertical angles are equal, so \( m\angle3 = m\angle2 = 98^\circ \)? Wait, no, vertical angles: \( \angle2 \) and \( \angle3 \)? Wait, no, let's check the diagram. Wait, \( \angle2 \) and \( \angle4 \) are supplementary? Wait, no, first, \( \angle2 \) and \( \angle3 \): Wait, maybe \( \angle2 \) and \( \angle4 \) are vertical? No, let's recall linear pairs and vertical angles. Also, lines \( c \) and \( d \) are parallel.

Step2: Check \( m\angle3 = m\angle6 = 98^\circ \)

If \( c \parallel d \), and a transversal cuts them, then corresponding angles or alternate interior angles? Wait, \( \angle3 \) and \( \angle6 \): if they are alternate interior angles, then \( m\angle3 = m\angle6 = 98^\circ \). So this statement is true.

Step3: Check \( m\angle3 = m\angle14 = 98^\circ \)

Wait, \( \angle3 \) and \( \angle14 \): are they related? Maybe not. Wait, let's check \( \angle4 \). Since \( \angle2 + \angle4 = 180^\circ \) (linear pair), so \( m\angle4 = 180^\circ - 98^\circ = 82^\circ \). Then, \( \angle4 \) and \( \angle8 \): if \( c \parallel d \), maybe alternate interior angles? So \( m\angle4 = m\angle8 = 82^\circ \). That's a true statement.

Step4: Check \( m\angle4 = m\angle8 = 82^\circ \)

As above, \( m\angle4 = 82^\circ \), and if \( c \parallel d \), \( \angle4 \) and \( \angle8 \) are alternate interior angles, so \( m\angle4 = m\angle8 = 82^\circ \). True.

Step5: Check \( m\angle5 = m\angle8 = 82^\circ \)

\( \angle5 \) and \( \angle8 \): vertical angles? Wait, \( \angle5 \) and \( \angle8 \) are vertical angles? So \( m\angle5 = m\angle8 \). Since \( m\angle8 = 82^\circ \), then \( m\angle5 = 82^\circ \), so \( m\angle5 = m\angle8 = 82^\circ \). True.

Wait, let's re-examine the options:

1. \( m\angle3 = m\angle6 = 98^\circ \): True (alternate interior angles, \( c \parallel d \))
2. \( m\angle3 = m\angle14 = 98^\circ \): Not sure, maybe false.
3. \( m\angle4 = m\angle8 = 82^\circ \): True (alternate interior angles, \( c \parallel d \), \( m\angle4 = 82^\circ \))
4. \( m\angle4 = m\angle12 = 82^\circ \): \( \angle4 \) and \( \angle12 \): are they related? Maybe not.
5. \( m\angle5 = m\angle8 = 82^\circ \): True (vertical angles, \( \angle5 \) and \( \angle8 \) are vertical, so equal, and \( m\angle8 = 82^\circ \))

Wait, the correct options should be:

• \( m\angle3 = m\angle6 = 98^\circ \)
• \( m\angle4 = m\angle8 = 82^\circ \)
• \( m\angle5 = m\angle8 = 82^\circ \)

Wait, let's confirm:

• \( \angle2 = 98^\circ \), so \( \angle3 = \angle2 = 98^\circ \) (vertical angles? Wait, no, \( \angle2 \) and \( \angle3 \): maybe \( \angle2 \) and \( \angle4 \) are vertical? Wait, maybe I messed up the diagram. But assuming the standard, the three true options are:

1. \( m\angle3 = m\angle6 = 98^\circ \)
2. \( m\angle4 = m\angle8 = 82^\circ \)
3. \( m\angle5 = m\angle8 = 82^\circ \)

Answer:

The three true options are:

• \( m\angle3 = m\angle6 = 98^\circ \)
• \( m\angle4 = m\angle8 = 82^\circ \)
• \( m\angle5 = m\angle8 = 82^\circ \)