Question

in the figure below, ( mangle1 = 64^{circ} ). find ( mangle2 ), ( mangle3 ), and ( mangle4 ).

Explanation:

Step1: Find \( m\angle2 \)

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

When two lines intersect, adjacent angles are supplementary (sum to \( 180^\circ \)), and vertical angles are equal.

So \( \angle1 \) and \( \angle2 \): Wait, no, \( \angle1 \) and \( \angle3 \) are vertical? Wait, the diagram: angles 1 and 3 are opposite (vertical), angles 2 and 4 are opposite (vertical). Angles 1 and 2 are adjacent, forming a linear pair.

So first, \( \angle1 \) and \( \angle2 \): are they adjacent? Let's see, angle 1 and angle 2 are adjacent, so they form a linear pair, so \( m\angle1 + m\angle2 = 180^\circ \).

Given \( m\angle1 = 64^\circ \), so \( m\angle2 = 180^\circ - 64^\circ = 116^\circ \)? Wait, no, wait, maybe I got the angles wrong. Wait, maybe \( \angle1 \) and \( \angle3 \) are vertical. Wait, let's re-examine the diagram:

The two lines intersect, creating four angles: 1, 2, 3, 4. Angle 1 is at the bottom, angle 2 at the right, angle 3 at the top, angle 4 at the left. So angle 1 and angle 3 are vertical angles (opposite each other), so \( m\angle1 = m\angle3 \). Angle 2 and angle 4 are vertical angles, so \( m\angle2 = m\angle4 \). Angle 1 and angle 2 are adjacent, forming a linear pair, so \( m\angle1 + m\angle2 = 180^\circ \).

Ah, that makes sense. So:

Step1: Find \( m\angle2 \)

\( \angle1 \) and \( \angle2 \) are supplementary (linear pair), so \( m\angle1 + m\angle2 = 180^\circ \).

Given \( m\angle1 = 64^\circ \), so \( m\angle2 = 180 - 64 = 116^\circ \).

Step2: Find \( m\angle3 \)

\( \angle1 \) and \( \angle3 \) are vertical angles, so \( m\angle3 = m\angle1 = 64^\circ \).

Step3: Find \( m\angle4 \)

\( \angle2 \) and \( \angle4 \) are vertical angles, so \( m\angle4 = m\angle2 = 116^\circ \). Wait, no, wait: \( \angle1 \) and \( \angle4 \) are adjacent? Wait, no, \( \angle1 \) and \( \angle4 \) are adjacent, forming a linear pair? Wait, no, let's correct:

Wait, angle 1 and angle 4: are they adjacent? Let's see the diagram: angle 1 is at the bottom, angle 4 is at the left, angle 2 at the right, angle 3 at the top. So angle 1 and angle 4 are adjacent, forming a linear pair? No, angle 1 and angle 2 are adjacent (linear pair), angle 2 and angle 3 (linear pair), angle 3 and angle 4 (linear pair), angle 4 and angle 1 (linear pair). And vertical angles: angle 1 and angle 3, angle 2 and angle 4.

So:

- Vertical angles: \( \angle1 \cong \angle3 \), \( \angle2 \cong \angle4 \)
- Linear pairs: \( \angle1 + \angle2 = 180^\circ \), \( \angle2 + \angle3 = 180^\circ \), \( \angle3 + \angle4 = 180^\circ \), \( \angle4 + \angle1 = 180^\circ \)

So:

\( m\angle3 = m\angle1 = 64^\circ \) (vertical angles)

\( m\angle2 = 180^\circ - m\angle1 = 180 - 64 = 116^\circ \) (linear pair with \( \angle1 \))

\( m\angle4 = m\angle2 = 116^\circ \) (vertical angles with \( \angle2 \))? Wait, no, \( \angle4 \) is vertical with \( \angle2 \)? Wait, no, \( \angle2 \) and \( \angle4 \) are vertical. So yes, \( m\angle4 = m\angle2 = 116^\circ \)? Wait, no, wait, \( \angle1 \) and \( \angle3 \) are vertical, so \( m\angle3 = 64^\circ \). \( \angle2 \) and \( \angle4 \) are vertical, so \( m\angle2 = m\angle4 \). And \( \angle1 \) and \( \angle2 \) are supplementary, so \( m\angle2 = 180 - 64 = 116 \), so \( m\angle4 = 116 \), and \( m\angle3 = 64 \).

Wait, let's verify:

- \( \angle1 = 64^\circ \)
- \( \angle2 \): adjacent to \( \angle1 \), linear pair, so \( 180 - 64 = 116^\circ \)
- \( \angle3 \): vertical to \( \angle1 \), so \( 64^\circ \)
- \( \angle4 \): vertical to \( \angle2 \), so \( 116^\circ \)

Yes, that makes sense.

Step1: Calculate \( m\angle2 \)

\( \angle1 \) and \( \angle2 \) are supplementary (linear pair), so:
\( m\angle2 = 180^\circ - m\angle1 \)
\( m\angle2 = 180^\circ - 64^\circ = 116^\circ \)

Step2: Calculate \( m\angle3 \)

\( \angle1 \) and \( \angle3 \) are vertical angles, so:
\( m\angle3 = m\angle1 = 64^\circ \)

Step3: Calculate \( m\angle4 \)

\( \angle2 \) and \( \angle4 \) are vertical angles, so:
\( m\angle4 = m\angle2 = 116^\circ \)

Answer:

\( m\angle2 = \boxed{116}^\circ \)
\( m\angle3 = \boxed{64}^\circ \)
\( m\angle4 = \boxed{116}^\circ \)