Question

finding angle measures
from one angle, a two - dimensional view
of a beach ball looks like a circle with two
chords intersecting, as shown. what are
the measures of each angle?
$m\angle2 + m\angle3 = 180^{\circ}$
$m\angle2 + 114^{\circ}=180^{\circ}$
$m\angle2 = 66^{\circ}$

Explanation:

Step1: Recall Vertical Angles and Linear Pairs

In a circle with intersecting chords, vertical angles are equal, and linear pairs of angles sum to \(180^\circ\). Angles \( \angle 1\) and \( \angle 3\) are vertical angles, \( \angle 2\) and \( \angle 4\) are vertical angles. Also, \( \angle 2\) and \( \angle 3\) form a linear pair (\(m\angle 2 + m\angle 3 = 180^\circ\)).

Step2: Find \(m\angle 2\)

Given \(m\angle 3 = 114^\circ\) (from the diagram, the red - marked angle), using the linear pair property \(m\angle 2+114^\circ = 180^\circ\).
Subtract \(114^\circ\) from both sides: \(m\angle 2=180^\circ - 114^\circ=66^\circ\).

Step3: Find \(m\angle 1\) and \(m\angle 4\)

Since \( \angle 1\) and \( \angle 3\) are vertical angles, \(m\angle 1 = m\angle 3 = 114^\circ\).
Since \( \angle 2\) and \( \angle 4\) are vertical angles, \(m\angle 4 = m\angle 2 = 66^\circ\).

Answer:

\(m\angle 1 = 114^\circ\), \(m\angle 2 = 66^\circ\), \(m\angle 3 = 114^\circ\), \(m\angle 4 = 66^\circ\)