Question

if \\(m\angle 2 = 78^\circ\\), which of the following angles also measure \\(78^\circ\\)? select three that apply.

a. \\(\angle 3\\)
b. \\(\angle 4\\)
c. \\(\angle 5\\)
d. \\(\angle 6\\)

Explanation:

Identify vertical angles

Using the Parallel Line Angles knowledge point, we first look at the angles formed at the intersection of line \(a\) and the transversal. Angles \(\angle 2\) and \(\angle 4\) are vertical angles because they are opposite each other. Since vertical angles are equal:

$$ m\angle 4 = m\angle 2 = 78^\circ $$

Apply corresponding angles

Using the Parallel Line Angles knowledge point, we relate the angles on line \(a\) to those on line \(b\). Since lines \(a\) and \(b\) are parallel, corresponding angles are equal. The angle corresponding to \(\angle 2\) is \(\angle 6\):

$$ m\angle 6 = m\angle 2 = 78^\circ $$

Apply alternate interior angles

Using the Alternate Interior Angles knowledge point, we relate the interior angles between the parallel lines. Angles \(\angle 4\) and \(\angle 6\) are alternate interior angles, which are equal. Additionally, we can find another angle equal to \(78^\circ\) by looking at alternate exterior angles or vertical angles on line \(b\). The angle vertical to \(\angle 6\) is \(\angle 8\), which is not listed. The angle alternate exterior to \(\angle 2\) is \(\angle 8\).

Let's check the remaining options:

• \(\angle 3\) is supplementary to \(\angle 2\): \(m\angle 3 = 180^\circ - 78^\circ = 102^\circ\).
• \(\angle 5\) is alternate interior to \(\angle 3\), so \(m\angle 5 = 102^\circ\).
• Let's re-evaluate the corresponding angle for \(\angle 4\), which is \(\angle 8\).
• The angle alternate interior to \(\angle 4\) is \(\angle 6\).
• The angle alternate exterior to \(\angle 2\) is \(\angle 8\).
• The angle corresponding to \(\angle 1\) is \(\angle 5\). Since \(\angle 1\) is vertical to \(\angle 3\), \(m\angle 1 = 102^\circ\), so \(m\angle 5 = 102^\circ\).
• Let's check the relationship for \(\angle 8\) and \(\angle 6\). Since \(\angle 6 = 78^\circ\), its vertical angle \(\angle 8 = 78^\circ\).
• Let's check \(\angle 4 = 78^\circ\).
• Let's check \(\angle 6 = 78^\circ\).
• Let's check \(\angle 8 = 78^\circ\) (not in options).
• Let's check \(\angle 1\) (not in options).
• Let's check if there is a third angle among the options. The options are A. \(\angle 3\), B. \(\angle 4\), C. \(\angle 5\), D. \(\angle 6\). Wait, the prompt says "Select THREE that apply", but only four options are shown: A, B, C, D. Let's check if there is an option E cut off or if we need to find which of these are \(78^\circ\). Among A, B, C, D:
• \(m\angle 3 = 102^\circ\)
• \(m\angle 4 = 78^\circ\)
• \(m\angle 5 = 102^\circ\)
• \(m\angle 6 = 78^\circ\)

Only two of the visible options (B and D) measure \(78^\circ\). The third one would be \(\angle 8\), which is likely option E (not visible in the cropped image). We will select the correct visible options.

Answer:

• A. \(\angle 3\)
• B. \(\angle 4\) (Correct answer)
• C. \(\angle 5\)
• D. \(\angle 6\) (Correct answer)