Question

in the figure below, m∠3 = 127°. find m∠1, m∠2, and m∠4.

Explanation:

Step1: Find \( m\angle1 \) (Vertical Angles)

Vertical angles are equal. \( \angle1 \) and \( \angle3 \) are vertical angles, so \( m\angle1 = m\angle3 \).
\( m\angle1 = 127^\circ \)

Step2: Find \( m\angle2 \) (Linear Pair)

\( \angle2 \) and \( \angle3 \) form a linear pair, so their sum is \( 180^\circ \). Let \( m\angle2 = x \), then \( x + 127^\circ = 180^\circ \). Solving for \( x \), we get \( x = 180^\circ - 127^\circ = 53^\circ \). So \( m\angle2 = 53^\circ \).

Step3: Find \( m\angle4 \) (Vertical Angles)

\( \angle4 \) and \( \angle2 \) are vertical angles (or \( \angle4 \) and \( \angle3 \) form a linear pair, either way works). Using vertical angles, \( m\angle4 = m\angle2 \). So \( m\angle4 = 53^\circ \) (or using linear pair with \( \angle1 \): \( 180^\circ - 127^\circ = 53^\circ \)).

Answer:

\( m\angle1 = 127^\circ \)
\( m\angle2 = 53^\circ \)
\( m\angle4 = 53^\circ \)