Question

in circle z, what is $m\angle2$?
$70^\circ$
$133^\circ$
$140^\circ$
$147^\circ$
circle z has arcs: $\overset{\frown}{ab}=133^\circ$, $\overset{\frown}{cd}=147^\circ$, with chords ad and bc intersecting inside the circle, forming $\angle1$ and $\angle2$

Explanation:

Step1: Find sum of known arcs

$133^\circ + 147^\circ = 280^\circ$

Step2: Find arc $\overset{\frown}{AD}$

$360^\circ - 280^\circ = 80^\circ$

Step3: Calculate $m\angle2$

$\frac{1}{2} \times (133^\circ + 80^\circ) = \frac{1}{2} \times 213^\circ$ Correction: Use inscribed angle rule for intersecting chords

Step3 (Revised): Apply intersecting chords angle formula

$m\angle2 = \frac{1}{2} \times (m\overset{\frown}{AB} + m\overset{\frown}{CD})$
$m\angle2 = \frac{1}{2} \times (133^\circ + (360^\circ - 133^\circ - 147^\circ))$
$m\angle2 = \frac{1}{2} \times (133^\circ + 80^\circ) = 106.5^\circ$ Correction: Match given options, use external angle rule

Step3 (Final): Use inscribed angle for intercepted arcs

$m\angle2 = \frac{1}{2} \times (m\overset{\frown}{AD} + m\overset{\frown}{BC})$
First find $m\overset{\frown}{AD} + m\overset{\frown}{BC} = 360^\circ - 133^\circ - 147^\circ = 80^\circ$ No, correct rule: angle formed by two chords = ½(sum of intercepted arcs)
$m\angle2 = \frac{1}{2} \times (133^\circ + 147^\circ)$ Incorrect, correct: angle intercepts arcs opposite

Step1: Calculate total of remaining arcs

$360^\circ - 133^\circ - 147^\circ = 80^\circ$

Step2: Apply intersecting chord angle theorem

$m\angle2 = \frac{1}{2} \times (133^\circ + 80^\circ) = 106.5^\circ$ *Discrepancy, re-evaluate: angle ∠2 intercepts arc AB (133°) and arc AD? No, correct: ∠2 is formed by chords AC and BD, so intercepted arcs are AB and CD. Arc CD = 360-133-147-arc BC? No, circle total 360: arc AB=133, arc CD=147, so arc BC + arc AD=80. ∠2 intercepts arc AB and arc AD? No, the correct formula is angle inside circle = ½(sum of intercepted arcs) = ½(arc AB + arc CD) = ½(133+147)=140°

Answer:

70° is incorrect, 133° is incorrect, 140°, 147° is incorrect.
Final Answer: $\boldsymbol{140^\circ}$ (Option C)