Question

consider the figure shown. classify each of the following statements as always true, sometimes true, or never true:
m∠1 + m∠4 = 180
m∠1 + m∠2 + m∠3 = 180
m∠2 + m∠4 = 180
∠2 ≅ ∠3
∠2 ≅ ∠4
m∠3 = m∠4

the statement m∠1 + m∠4 = 180 is dropdown
the statement m∠1 + m∠2 + m∠3 = 180 is dropdown
the statement m∠2 + m∠4 = 180 is dropdown
the statement ∠2 ≅ ∠3 is dropdown

Explanation:

Step1: Analyze \( m\angle1 + m\angle4 = 180 \)

From the figure, \( \angle1 \) and \( \angle4 \) are adjacent and form a linear pair? Wait, no, looking at the intersection, \( \angle1 \) and \( \angle4 \): actually, \( \angle1 \) and \( \angle2 \) are linear pair, \( \angle1 \) and \( \angle3 \) are vertical angles? Wait, no, the two lines intersect, so \( \angle1 \) and \( \angle3 \) are vertical, \( \angle2 \) and \( \angle4 \) are vertical. \( \angle1 \) and \( \angle2 \) are supplementary (linear pair), \( \angle1 \) and \( \angle4 \): let's see, \( \angle1 \) and \( \angle4 \): are they adjacent? Wait, the angles around a point sum to 360, but linear pairs sum to 180. Wait, \( \angle1 \) and \( \angle4 \): if we look at the intersection, \( \angle1 \) and \( \angle4 \) are adjacent? Wait, no, \( \angle1 \) is between \( \angle2 \) and \( \angle4 \)? Wait, the figure shows two intersecting lines, so four angles: \( \angle1 \), \( \angle2 \), \( \angle3 \), \( \angle4 \), with \( \angle1 \) and \( \angle3 \) vertical, \( \angle2 \) and \( \angle4 \) vertical. \( \angle1 \) and \( \angle2 \) are supplementary (linear pair, sum to 180), \( \angle1 \) and \( \angle4 \): are they supplementary? Wait, \( \angle1 + \angle2 = 180 \), \( \angle2 = \angle4 \) (vertical angles), so \( \angle1 + \angle4 = 180 \)? Wait, no, \( \angle2 \) and \( \angle4 \) are vertical, so \( \angle2 = \angle4 \). \( \angle1 + \angle2 = 180 \), so \( \angle1 + \angle4 = 180 \) (since \( \angle2 = \angle4 \))? Wait, no, that would be if \( \angle2 = \angle4 \), but \( \angle1 + \angle2 = 180 \), so \( \angle1 + \angle4 = 180 \) only if \( \angle2 = \angle4 \), which they are (vertical angles). Wait, no, vertical angles are equal, so \( \angle2 = \angle4 \), \( \angle1 = \angle3 \). So \( \angle1 + \angle2 = 180 \) (linear pair), so \( \angle1 + \angle4 = 180 \) (since \( \angle2 = \angle4 \))? Wait, no, that's not right. Wait, \( \angle1 \) and \( \angle4 \): are they adjacent? Let's visualize: two lines intersect, so angle 1, angle 2 (above angle 1), angle 3 (opposite angle 1), angle 4 (below angle 1). So angle 1 and angle 4: are they adjacent? Yes, they share a side and a vertex. So angle 1 and angle 4: do they form a linear pair? Wait, a linear pair is two adjacent angles that form a straight line. So angle 1 and angle 2 form a straight line (linear pair), angle 1 and angle 4: do they form a straight line? No, because angle 1 is between angle 2 and angle 4, so angle 1 and angle 4 are adjacent but not a linear pair? Wait, no, the sum of angles around a point is 360, so \( \angle1 + \angle2 + \angle3 + \angle4 = 360 \). But \( \angle1 = \angle3 \), \( \angle2 = \angle4 \), so \( 2\angle1 + 2\angle2 = 360 \), so \( \angle1 + \angle2 = 180 \), which is linear pair. Now, \( \angle1 + \angle4 \): since \( \angle4 = \angle2 \), then \( \angle1 + \angle4 = \angle1 + \angle2 = 180 \). So \( m\angle1 + m\angle4 = 180 \) is always true? Wait, no, wait, maybe I made a mistake. Wait, \( \angle1 \) and \( \angle4 \): if the two lines are intersecting, then \( \angle1 \) and \( \angle4 \) are adjacent and form a linear pair? Wait, no, linear pair is two angles that are adjacent and their non - common sides form a straight line. So \( \angle1 \) and \( \angle2 \): non - common sides form a straight line. \( \angle1 \) and \( \angle4 \): non - common sides? Let's see, angle 1 has sides: one from the first line, one from the second line. Angle 4 has sides: same two lines, but opposite. Wait, maybe \( \angle1 \) and \( \angle4 \) are supplementary. So \( m\angle1 + m\angle4 = 180 \) is always true? Wait, no, wait, vertical angles are equal, linear pairs are supplementary. Wait, \( \angle1 \) and \( \angle2 \): supplementary (linear pair), \( \angle2 \) and \( \angle3 \): supplementary (linear pair), \( \angle3 \) and \( \angle4 \): supplementary (linear pair), \( \angle4 \) and \( \angle1 \): supplementary (linear pair)? Wait, no, that can't be. Wait, no, when two lines intersect, there are two pairs of vertical angles and four linear pairs. Wait, each angle forms a linear pair with two adjacent angles. So \( \angle1 \) forms a linear pair with \( \angle2 \) and \( \angle4 \). So \( \angle1 + \angle2 = 180 \) and \( \angle1 + \angle4 = 180 \). So \( m\angle1 + m\angle4 = 180 \) is always true.

Step2: Analyze \( m\angle1 + m\angle2 + m\angle3 = 180 \)

We know that \( \angle1 + \angle2 = 180 \) (linear pair), and \( \angle3=\angle1 \) (vertical angles). So \( m\angle1 + m\angle2 + m\angle3=\angle1+\angle2+\angle1 = 180+\angle1 \). Since \( \angle1 \) is not necessarily 0, this sum is \( 180 + \angle1 \), which is greater than 180 unless \( \angle1 = 0 \), which it's not. Wait, no, wait, \( \angle1 \) and \( \angle3 \) are vertical angles, so \( \angle3=\angle1 \). \( \angle1 + \angle2 = 180 \), so \( \angle1 + \angle2 + \angle3=\angle1+\angle2+\angle1 = 2\angle1 + \angle2 \). But \( \angle1 + \angle2 = 180 \), so substitute \( \angle2 = 180 - \angle1 \), then the sum is \( 2\angle1+(180 - \angle1)=180+\angle1 \). Since \( \angle1 \) is a non - zero angle (it's formed by two intersecting lines), \( 180+\angle1>180 \). Wait, but maybe I misread the angles. Wait, the figure: maybe \( \angle1 \), \( \angle2 \), \( \angle3 \) are angles in a triangle? No, the figure is two intersecting lines. Wait, no, maybe the angles are \( \angle1 \), \( \angle2 \), \( \angle3 \) where \( \angle1 \) and \( \angle2 \) are linear pair, and \( \angle3 \) is adjacent? Wait, no, the problem says "the figure shown" with two intersecting lines, so angles 1,2,3,4. So \( \angle1 + \angle2 + \angle3 \): \( \angle1 \) and \( \angle2 \) sum to 180, \( \angle3 \) is equal to \( \angle1 \) (vertical), so sum is \( 180+\angle1 \), which is more than 180. But wait, maybe the figure is different? Wait, maybe it's a triangle? No, the problem says two intersecting lines. Wait, maybe I made a mistake. Wait, no, if we have two intersecting lines, the sum of \( \angle1 + \angle2 + \angle3 \): \( \angle1 \) and \( \angle2 \) are supplementary (180), \( \angle3 \) is equal to \( \angle1 \) (vertical), so sum is 180 + \( \angle1 \), which is always greater than 180, so \( m\angle1 + m\angle2 + m\angle3 = 180 \) is never true? Wait, no, wait, maybe the angles are \( \angle1 \), \( \angle2 \), \( \angle3 \) in a straight line? No, two intersecting lines form four angles. Wait, maybe the problem has a typo, but based on the figure (two intersecting lines), \( \angle1 + \angle2 + \angle3 \): \( \angle1 \) and \( \angle2 \) are linear pair (180), \( \angle3 \) is vertical to \( \angle1 \), so \( \angle3=\angle1 \), so sum is 180 + \( \angle1 \), which is more than 180, so the statement is never true? Wait, no, maybe I misinterpret the angles. Wait, maybe \( \angle1 \), \( \angle2 \), \( \angle3 \) are angles around a point? No, around a point it's 360. Wait, no, the problem says "classify each statement". Let's re - examine: \( m\angle1 + m\angle2 + m\angle3 = 180 \). If \( \angle1 \), \( \angle2 \), \( \angle3 \) are angles in a triangle, but the figure is two intersecting lines. Wait, maybe the figure is a triangle with a transversal? No, the problem says two intersecting lines. So I think the sum \( \angle1 + \angle2 + \angle3 \) is \( 180+\angle1 \) (since \( \angle3=\angle1 \)), so it's never equal to 180, so the statement is never true? Wait, no, maybe \( \angle1 \) is 0, which it's not. So \( m\angle1 + m\angle2 + m\angle3 = 180 \) is never true.

Step3: Analyze \( m\angle2 + m\angle4 = 180 \)

\( \angle2 \) and \( \angle4 \) are vertical angles, so \( \angle2=\angle4 \). So \( m\angle2 + m\angle4 = 2m\angle2 \). For this to be 180, \( m\angle2 = 90 \), i.e., the lines are perpendicular. But the figure doesn't show perpendicular lines, so this is sometimes true (when the lines are perpendicular) and sometimes false (when they are not).

Step4: Analyze \( \angle2\cong\angle3 \)

\( \angle2 \) and \( \angle3 \): are they congruent? \( \angle2 \) and \( \angle1 \) are supplementary, \( \angle3 \) and \( \angle1 \) are vertical (so \( \angle3=\angle1 \)). \( \angle2 \) and \( \angle3 \): \( \angle2 = 180 - \angle1 \), \( \angle3=\angle1 \). So \( \angle2\cong\angle3 \) only if \( 180 - \angle1=\angle1 \), i.e., \( \angle1 = 90 \), so the lines are perpendicular. So this is sometimes true (when lines are perpendicular) and sometimes false (when not).

Answer:

• The statement \( m\angle1 + m\angle4 = 180 \) is always true.
• The statement \( m\angle1 + m\angle2 + m\angle3 = 180 \) is never true.
• The statement \( m\angle2 + m\angle4 = 180 \) is sometimes true.
• The statement \( \angle2\cong\angle3 \) is sometimes true.