Question

1. name a pair of consecutive interior and exterior angles, corresponding angles, alternative interior angles, alternative exterior angles, vertical angles, and a pair of linear pairs. if \\(m\angle 3\\) is 78, what is the \\(m\angle 7\\)? if \\(m\angle 4\\) is 25, what is \\(m\angle 6\\)?

Explanation:

Identify angle relationships

Using the Consecutive Interior Angles, Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, and Vertical Angles Theorem knowledge points

• Consecutive interior: \(\angle 3\) and \(\angle 5\), or \(\angle 4\) and \(\angle 6\)
• Consecutive exterior: \(\angle 1\) and \(\angle 7\), or \(\angle 2\) and \(\angle 8\)
• Corresponding: \(\angle 1\) and \(\angle 5\), \(\angle 2\) and \(\angle 6\), \(\angle 3\) and \(\angle 7\), or \(\angle 4\) and \(\angle 8\)
• Alternate interior: \(\angle 3\) and \(\angle 6\), or \(\angle 4\) and \(\angle 5\)
• Alternate exterior: \(\angle 1\) and \(\angle 8\), or \(\angle 2\) and \(\angle 7\)
• Vertical: \(\angle 1\) and \(\angle 4\), \(\angle 2\) and \(\angle 3\), \(\angle 5\) and \(\angle 8\), or \(\angle 6\) and \(\angle 7\)
• Linear pairs: \(\angle 1\) and \(\angle 2\), \(\angle 3\) and \(\angle 4\), \(\angle 5\) and \(\angle 6\), or \(\angle 7\) and \(\angle 8\)

Calculate measure of angle 7

Using the Alternate Exterior Angles and Vertical Angles Theorem knowledge points

• Given: \(m\angle 3 = 78^\circ\)
• \(\angle 2\) and \(\angle 3\) are vertical angles, so \(m\angle 2 = m\angle 3 = 78^\circ\)
• \(\angle 2\) and \(\angle 7\) are alternate exterior angles, so \(m\angle 7 = m\angle 2 = 78^\circ\)

Calculate measure of angle 6

Using the Consecutive Interior Angles knowledge point

• Given: \(m\angle 4 = 25^\circ\)
• \(\angle 4\) and \(\angle 6\) are consecutive interior angles, which are supplementary:

$$ m\angle 4 + m\angle 6 = 180^\circ $$

$$ 25^\circ + m\angle 6 = 180^\circ \implies m\angle 6 = 155^\circ $$

Answer:

• Consecutive interior angles: \(\angle 3\) and \(\angle 5\) (or \(\angle 4\) and \(\angle 6\))
• Consecutive exterior angles: \(\angle 1\) and \(\angle 7\) (or \(\angle 2\) and \(\angle 8\))
• Corresponding angles: \(\angle 1\) and \(\angle 5\) (or \(\angle 2\) and \(\angle 6\), \(\angle 3\) and \(\angle 7\), \(\angle 4\) and \(\angle 8\))
• Alternate interior angles: \(\angle 3\) and \(\angle 6\) (or \(\angle 4\) and \(\angle 5\))
• Alternate exterior angles: \(\angle 1\) and \(\angle 8\) (or \(\angle 2\) and \(\angle 7\))
• Vertical angles: \(\angle 1\) and \(\angle 4\) (or \(\angle 2\) and \(\angle 3\), \(\angle 5\) and \(\angle 8\), \(\angle 6\) and \(\angle 7\))
• Linear pairs: \(\angle 1\) and \(\angle 2\) (or any adjacent angles sharing a straight line)

• If \(m\angle 3 = 78^\circ\), then \(m\angle 7 = 78^\circ\)
• If \(m\angle 4 = 25^\circ\), then \(m\angle 6 = 155^\circ\)