Question

in the figure below, m∠1 = 128°. find m∠2, m∠3, and m∠4.

m∠2 =

m∠3 =

m∠4 =

Explanation:

Step1: Find \( m\angle2 \) (Vertical Angles)

Vertical angles are equal. \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, \( \angle1 \) and \( \angle3 \)? Wait, no, \( \angle1 \) and \( \angle3 \) are vertical? Wait, no, \( \angle1 \) and \( \angle3 \) are vertical? Wait, no, when two lines intersect, vertical angles are opposite. So \( \angle1 \) and \( \angle3 \) are vertical? Wait, no, \( \angle1 \) and \( \angle3 \) are adjacent to \( \angle2 \) and \( \angle4 \). Wait, actually, \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, let's correct. When two lines intersect, the vertical angles are equal. So \( \angle1 \) and \( \angle3 \) are vertical? Wait, no, \( \angle1 \) and \( \angle3 \) are adjacent to \( \angle2 \). Wait, \( \angle1 \) and \( \angle2 \) are supplementary because they form a linear pair. So \( m\angle1 + m\angle2 = 180^\circ \). Given \( m\angle1 = 128^\circ \), so \( m\angle2 = 180^\circ - 128^\circ = 52^\circ \).

Step2: Find \( m\angle3 \) (Vertical Angles with \( \angle1 \))

Vertical angles are equal. \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, \( \angle1 \) and \( \angle3 \) are vertical? Wait, no, \( \angle1 \) and \( \angle3 \) are opposite? Wait, when two lines intersect, the vertical angles are opposite. So \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, \( \angle1 \) and \( \angle3 \) are adjacent to \( \angle2 \) and \( \angle4 \). Wait, actually, \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, \( \angle1 \) and \( \angle3 \) are supplementary? No, wait, \( \angle1 \) and \( \angle2 \) are supplementary, \( \angle2 \) and \( \angle3 \) are supplementary? No, wait, \( \angle1 \) and \( \angle3 \) are vertical angles, so they are equal. Wait, no, let's draw the intersecting lines. So two lines intersect, forming four angles: \( \angle1 \), \( \angle2 \), \( \angle3 \), \( \angle4 \). \( \angle1 \) and \( \angle3 \) are vertical angles (opposite each other), \( \angle2 \) and \( \angle4 \) are vertical angles. \( \angle1 \) and \( \angle2 \) are adjacent, forming a linear pair (supplementary). So \( m\angle1 + m\angle2 = 180^\circ \), \( m\angle2 + m\angle3 = 180^\circ \), so \( m\angle1 = m\angle3 \) (vertical angles). Wait, no, \( m\angle1 = 128^\circ \), so \( m\angle3 = 128^\circ \) because they are vertical angles. Wait, that's a mistake earlier. Wait, \( \angle1 \) and \( \angle3 \) are vertical angles, so they are equal. \( \angle2 \) and \( \angle4 \) are vertical angles, so they are equal. And \( \angle1 \) and \( \angle2 \) are supplementary (linear pair), so \( m\angle1 + m\angle2 = 180^\circ \), so \( m\angle2 = 52^\circ \), then \( m\angle4 = m\angle2 = 52^\circ \), and \( m\angle3 = m\angle1 = 128^\circ \).

Step3: Find \( m\angle4 \) (Vertical Angles with \( \angle2 \))

Vertical angles are equal. \( \angle2 \) and \( \angle4 \) are vertical angles, so \( m\angle4 = m\angle2 = 52^\circ \).

Answer:

\( m\angle2 = 52^\circ \), \( m\angle3 = 128^\circ \), \( m\angle4 = 52^\circ \)