Question

in the figure below, $m\angle 4 = 96^{\circ}$. find $m\angle 1$, $m\angle 2$, and $m\angle 3$.
$m\angle 1 = \square^{\circ}$
$m\angle 2 = \square^{\circ}$
$m\angle 3 = \square^{\circ}$

Explanation:

Step1: Find \( m\angle1 \)

\( \angle1 \) and \( \angle4 \) are adjacent supplementary angles? No, wait, actually \( \angle1 \) and \( \angle4 \) are adjacent and form a linear pair? Wait, no, looking at the intersecting lines, \( \angle1 \) and \( \angle3 \) are vertical angles, \( \angle2 \) and \( \angle4 \) are vertical angles, and \( \angle1 \) and \( \angle4 \) are supplementary (since they form a linear pair, sum to \( 180^\circ \)). Wait, let's correct: when two lines intersect, vertical angles are equal, and adjacent angles (linear pair) are supplementary (sum to \( 180^\circ \)).

So \( \angle1 \) and \( \angle4 \) are supplementary? Wait, no, \( \angle1 \) and \( \angle4 \): let's see the figure. The angles around the intersection: \( \angle1 \), \( \angle2 \), \( \angle3 \), \( \angle4 \). So \( \angle1 \) and \( \angle3 \) are vertical angles (equal), \( \angle2 \) and \( \angle4 \) are vertical angles (equal). \( \angle1 \) and \( \angle2 \) are supplementary, \( \angle2 \) and \( \angle3 \) are supplementary, etc.

Given \( m\angle4 = 96^\circ \).

First, \( \angle1 \) and \( \angle4 \): are they adjacent? Wait, maybe I mislabel. Let's see: the two lines intersect, so \( \angle1 \) and \( \angle4 \) are adjacent and form a linear pair? Wait, no, if \( \angle4 \) is at the bottom, \( \angle1 \) is at the right, \( \angle2 \) at the top, \( \angle3 \) at the left. So \( \angle1 \) and \( \angle4 \) are adjacent, forming a linear pair, so \( m\angle1 + m\angle4 = 180^\circ \).

So \( m\angle1 = 180^\circ - m\angle4 = 180^\circ - 96^\circ = 84^\circ \).

Step2: Find \( m\angle2 \)

\( \angle2 \) and \( \angle4 \) are vertical angles? Wait, no, vertical angles are opposite each other. Wait, \( \angle2 \) and \( \angle4 \): if \( \angle4 \) is at the bottom, \( \angle2 \) is at the top, so they are vertical angles? Wait, no, vertical angles are \( \angle1 \) and \( \angle3 \), \( \angle2 \) and \( \angle4 \)? Wait, no, when two lines intersect, the vertical angles are the pairs opposite each other. So if the angles are labeled in order around the intersection: \( \angle1 \) (right), \( \angle2 \) (top), \( \angle3 \) (left), \( \angle4 \) (bottom). Then \( \angle1 \) and \( \angle3 \) are vertical (opposite), \( \angle2 \) and \( \angle4 \) are vertical (opposite). So vertical angles are equal. So \( m\angle2 = m\angle4 = 96^\circ \)? Wait, no, that can't be, because \( \angle1 \) and \( \angle2 \) should be supplementary. Wait, maybe I got the labels wrong.

Wait, let's start over. When two lines intersect, the sum of adjacent angles (linear pair) is \( 180^\circ \). So \( \angle1 \) and \( \angle4 \): are they adjacent? Let's see the figure: the two lines cross, so \( \angle1 \) is adjacent to \( \angle4 \) and \( \angle2 \). So \( \angle1 + \angle4 = 180^\circ \) (linear pair). So \( m\angle1 = 180 - 96 = 84^\circ \).

Then \( \angle1 \) and \( \angle3 \) are vertical angles, so \( m\angle3 = m\angle1 = 84^\circ \).

\( \angle2 \) and \( \angle4 \) are vertical angles? Wait, no, \( \angle2 \) and \( \angle4 \): if \( \angle2 \) is adjacent to \( \angle1 \) and \( \angle3 \), then \( \angle2 + \angle1 = 180^\circ \), so \( m\angle2 = 180 - 84 = 96^\circ \), which is equal to \( m\angle4 \). So vertical angles: \( \angle2 \) and \( \angle4 \) are vertical (equal), \( \angle1 \) and \( \angle3 \) are vertical (equal).

So:

- \( \angle1 \) and \( \angle4 \) are supplementary: \( m\angle1 + m\angle4 = 180^\circ \)
- \( \angle2 \) and \( \angle4 \) are vertical angles: \( m\angle2 = m\angle4 \)
- \( \angle1 \) and \( \angle3 \) are vertical angles: \( m\angle3 = m\angle1 \)

So:

Step1: Calculate \( m\angle1 \)

\( m\angle1 = 180^\circ - m\angle4 = 180 - 96 = 84^\circ \)

Step2: Calculate \( m\angle2 \)

\( m\angle2 = m\angle4 = 96^\circ \) (vertical angles are equal)

Step3: Calculate \( m\angle3 \)

\( m\angle3 = m\angle1 = 84^\circ \) (vertical angles are equal)

Answer:

\( m\angle1 = \boxed{84}^\circ \)

\( m\angle2 = \boxed{96}^\circ \)

\( m\angle3 = \boxed{84}^\circ \)