in $ \odot o, m \widehat{bd}=120^{\circ} $ and $ \overline{ab} \cong \overline{cd} $. also, the center of the circle, point $ o $, is the intersection of $ \overline{cb} $ and $ \overline{ad} $. what is $ m \angle 4 $? $ m \angle 4=\square^{\circ} $

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Accepted Answer

# Explanation:
## Step1: Use the property of congruent chords
Since $\overline{AB}\cong\overline{CD}$, then the central angles subtended by these chords are equal. Let $\angle2 = \angle1$.
## Step2: Calculate the measure of $\angle BOD$
We know that $m\widehat{BD}=120^{\circ}$, and the central angle $\angle BOD$ has the same measure as the arc $\widehat{BD}$, so $m\angle BOD = 120^{\circ}$.
## Step3: Use the angle - sum property of a circle ($360^{\circ}$)
Since $\angle1+\angle2+\angle BOD = 180^{\circ}$ (linear pair adjacent to the diameter - related angles in a circle, sum of angles around a straight line in the circle's context). Let $x=\angle1=\angle2$, then $2x + 120^{\circ}=180^{\circ}$. Solving for $x$:
$$
\begin{align*}
2x&=180^{\circ}- 120^{\circ}\
2x&=60^{\circ}\
x& = 30^{\circ}
\end{align*}
$$
## Step4: Use the property of inscribed angles subtended by the same arc
$\angle4$ and $\angle1$ are angles in $\triangle OCD$ and $\triangle OAB$ (where $OA = OB=OC = OD$ as radii). Also, $\angle4$ and $\angle1$ can be found using the property that $\angle4=\angle1$ (alternate - interior - like angles in the circle's isosceles - triangle - based angle relationships).

# Answer:
$30$

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QUESTION IMAGE

Question

Explanation:

Step1: Use the property of congruent chords

Step2: Calculate the measure of \(\angle BOD\)

Step3: Use the angle - sum property of a circle (\(360^{\circ}\))

Step4: Use the property of inscribed angles subtended by the same arc

Answer: