Question

in the figure below, m∠2 = 55°, find m∠1, m∠3, and m∠4

Explanation:

Step1: Find \( m\angle1 \) (supplementary angles)

\(\angle1\) and \(\angle2\) are supplementary, so \( m\angle1 + m\angle2 = 180^\circ \). Given \( m\angle2 = 55^\circ \), then \( m\angle1 = 180^\circ - 55^\circ = 125^\circ \).

Step2: Find \( m\angle3 \) (vertical angles)

\(\angle3\) and \(\angle1\) are vertical angles? No, \(\angle3\) and \(\angle1\)? Wait, \(\angle2\) and \(\angle3\)? Wait, \(\angle1\) and \(\angle3\) are vertical? No, \(\angle2\) and \(\angle3\) are supplementary? Wait, no, vertical angles: \(\angle1\) and \(\angle3\) are vertical? Wait, the figure: two intersecting lines, so \(\angle1\) and \(\angle3\) are vertical? No, \(\angle1\) and \(\angle3\) – wait, \(\angle2\) and \(\angle4\) are vertical, \(\angle1\) and \(\angle3\) are vertical? Wait, no, when two lines intersect, vertical angles are equal. So \(\angle1\) and \(\angle3\) – wait, \(\angle2\) and \(\angle4\) are vertical, \(\angle1\) and \(\angle3\) are vertical. Wait, \(\angle1\) and \(\angle2\) are supplementary, \(\angle2\) and \(\angle3\) are supplementary? No, \(\angle2\) and \(\angle3\) are adjacent? Wait, no, let's correct: when two lines intersect, adjacent angles are supplementary, vertical angles are equal. So \(\angle1\) and \(\angle2\) are adjacent (supplementary), \(\angle2\) and \(\angle3\) are adjacent (supplementary)? No, \(\angle1\) and \(\angle3\) are vertical angles (equal), \(\angle2\) and \(\angle4\) are vertical angles (equal). Wait, no, let's re-express:

If two lines intersect, then:

- Adjacent angles (forming a linear pair) are supplementary (sum to \( 180^\circ \)).

- Vertical angles are equal.

So, \(\angle1\) and \(\angle2\) form a linear pair (supplementary), so \( m\angle1 = 180^\circ - 55^\circ = 125^\circ \).

\(\angle2\) and \(\angle3\) form a linear pair? No, \(\angle2\) and \(\angle3\) – wait, the labels: let's assume the angles are labeled 1, 2, 3, 4 around the intersection. So angle 1 and angle 2 are adjacent (linear pair), angle 2 and angle 3 are adjacent (linear pair), angle 3 and angle 4, angle 4 and angle 1. So vertical angles: angle 1 and angle 3, angle 2 and angle 4.

So, \( m\angle3 = m\angle1 = 125^\circ \)? No, wait, no: angle 2 and angle 3 are adjacent, so \( m\angle2 + m\angle3 = 180^\circ \)? No, that can't be. Wait, no, I think I mixed up. Let's take the standard: when two lines intersect, the vertical angles are equal, and adjacent angles (linear pair) are supplementary.

So, if \( m\angle2 = 55^\circ \), then:

- \(\angle1\) and \(\angle2\) are linear pair: \( m\angle1 = 180^\circ - 55^\circ = 125^\circ \).

- \(\angle2\) and \(\angle3\) are vertical angles? No, \(\angle2\) and \(\angle4\) are vertical. Wait, maybe the labels are: angle 2, angle 1, angle 4, angle 3 around the intersection. So angle 2 and angle 4 are vertical, angle 1 and angle 3 are vertical. Then angle 2 and angle 1 are linear pair, angle 1 and angle 4 are linear pair, etc.

So, \(\angle2\) (55°) and \(\angle4\) are vertical, so \( m\angle4 = 55^\circ \).

\(\angle1\) and \(\angle2\) are linear pair, so \( m\angle1 = 180 - 55 = 125^\circ \).

\(\angle1\) and \(\angle3\) are vertical, so \( m\angle3 = 125^\circ \).

Wait, the problem says "find \( m\angle1 \), \( m\angle3 \), and \( m\angle4 \)".

So:

Step1: \( m\angle1 \) (linear pair with \(\angle2\))

\( m\angle1 + m\angle2 = 180^\circ \)
\( m\angle1 = 180^\circ - 55^\circ = 125^\circ \)

Step2: \( m\angle3 \) (vertical with \(\angle1\))

Vertical angles are equal, so \( m\angle3 = m\angle1 = 125^\circ \)

Step3: \( m\angle4 \) (vertical with \(\angle2\))

Vertical angles are equal, so \( m\angle4 = m\angle2 = 55^\circ \)

Answer:

\( m\angle1 = 125^\circ \), \( m\angle3 = 125^\circ \), \( m\angle4 = 55^\circ \)