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In the diagram below of circle
O, chords
bar(AB) and
bar(CD) intersect at
E.
If
m/_AEC=34 and
mAC^(⏜)=50, what is
mDB^(⏜) ?

In the diagram below of circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ intersect at $E$ . $\newline$ If $\mathrm{m} \angle A E C=34$ and \mathrm{m} \overparen{A C}=50 , what is \mathrm{m} \overparen{D B} ?

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. In the diagram below of circle

O

, chords

\overline{A B}

and

\overline{C D}

intersect at

E

.

\newline

If

\mathrm{m} \angle A E C=34

and \mathrm{m} \overparen{A C}=50 , what is \mathrm{m} \overparen{D B} ?

Vertical Angles Congruence: Since $AE$ and $CD$ intersect at $E$ inside the circle, angles $\angle AEC$ and $\angle DEB$ are vertical angles and therefore congruent. $\newline$ $m/\angle DEB = m/\angle AEC = 34^\circ$ .
Sum of Angles in Same Segment: The angles $AEC$ and $ACB$ are part of the same segment, so their measures add up to $mAC^{\bigcirc}$ . $\angle ACB = mAC^{\bigcirc} - \angle AEC = 50^\circ - 34^\circ = 16^\circ$ .
Inscribed Angle Property: The angle $ACB$ is inscribed in the arc $ABC$ , so the measure of arc $ABC$ is twice the measure of angle $ACB$ . $\newline$ $m_{AB}^{\bigcirc} = 2 \times m_{\angle ACB} = 2 \times 16^\circ = 32^\circ$ .
Rest of the Circle: The full circle is $360$ degrees, so the measure of arc $DB$ is the rest of the circle minus arc $ABC$ . $\newline$ $m_{DB}^{\bigcirc} = 360$ degrees $- m_{AB}^{\bigcirc} = 360$ degrees $- 32$ degrees $= 328$ degrees.

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