Question

in circle z, what is m∠2? 70° 133° 140° 147°

Explanation:

Step1: Find the measure of the arc opposite to ∠1

The sum of arcs in a circle is \(360^{\circ}\). Let the arc opposite to \(\angle1\) be \(x\). Then \(x = 360^{\circ}-133^{\circ}-147^{\circ}\)

$$x=360^{\circ}-(133^{\circ} + 147^{\circ})=360^{\circ}-280^{\circ}=80^{\circ}$$

Step2: Use the property of vertical angles and inscribed angles

\(\angle1\) is an inscribed angle. The measure of an inscribed angle is half the measure of its intercepted arc. So \(m\angle1=\frac{1}{2}\times80^{\circ} = 40^{\circ}\)
Since \(\angle1\) and \(\angle2\) are vertical angles, \(m\angle1 + m\angle2=180^{\circ}\) (linear - pair adjacent angles in intersecting chords, or we can use the property that the sum of angles around a point on a straight line formed by intersecting chords). So \(m\angle2=180^{\circ}-m\angle1\)
Substitute \(m\angle1 = 40^{\circ}\) into the equation: \(m\angle2=180^{\circ}-40^{\circ}=140^{\circ}\)

Answer:

\(140^{\circ}\) (the third option)