Question

in the figure below, $m\angle 1 = 71^{\circ}$. find $m\angle 2$, $m\angle 3$, and $m\angle 4$.

Explanation:

Step1: Find \( m\angle2 \) (supplementary to \( \angle1 \))

\( \angle1 \) and \( \angle2 \) are supplementary (form a linear pair), so \( m\angle1 + m\angle2 = 180^\circ \).
\( 71^\circ + m\angle2 = 180^\circ \)
\( m\angle2 = 180^\circ - 71^\circ = 109^\circ \)

Step2: Find \( m\angle3 \) (vertical to \( \angle1 \))

\( \angle1 \) and \( \angle3 \) are vertical angles, so they are equal.
\( m\angle3 = m\angle1 = 71^\circ \)

Step3: Find \( m\angle4 \) (vertical to \( \angle2 \))

\( \angle2 \) and \( \angle4 \) are vertical angles, so they are equal.
\( m\angle4 = m\angle2 = 109^\circ \)

Answer:

\( m\angle2 = \boldsymbol{109}^\circ \)
\( m\angle3 = \boldsymbol{71}^\circ \)
\( m\angle4 = \boldsymbol{109}^\circ \)