Question

marcel is designing a circular necklace that will consist of 4 sections, each with a different color of plastic. determine how he needs to cut the plastic by finding the measures of the angles. m∠1 = (square^{circ}) m∠2 = (square^{circ})

Explanation:

Step1: Recall angle - sum property of a circle

The sum of angles around a point is 360°.

Step2: Set up an equation for the given angles

Let the four angles be 54°, 112°, \(m\angle1\), and \(m\angle2\). We know that \(54 + 112+m\angle1 + m\angle2=360\). Also, since the lines are intersecting at the center of the circle, vertical - angles are equal. Let's assume the circle is divided into four non - overlapping central angles.
We know that vertical angles are equal. So, if we consider the pairs of vertical angles, we can find the angles.
We know that \(m\angle1\) and the angle opposite to the 54° angle are vertical angles, and \(m\angle2\) and the angle opposite to the 112° angle are vertical angles.
The sum of the two given non - vertical angles is \(54 + 112 = 166\).
The sum of the other two angles \(m\angle1+m\angle2=360-(54 + 112)=194\).
Since vertical angles are equal, \(m\angle1\) and the angle opposite to 54° are equal, and \(m\angle2\) and the angle opposite to 112° are equal.
We know that \(m\angle1\) and the angle adjacent to it (112°) and \(m\angle2\) and the angle adjacent to it (54°) form the full - circle.
Since vertical angles are equal, \(m\angle1 = 54^{\circ}\) and \(m\angle2=112^{\circ}\)

Answer:

\(m\angle1 = 54\)
\(m\angle2 = 112\)