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Posted 2 months ago

A circle is centered on point BB. Points A,CA, C and DD lie on its circumference.
If ABC\angle A B C measures 124124^{\circ}, what does ADC\angle A D C measure?
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
Identify the relationship between the central angle and the inscribed angle that intercepts the same arc
step 2
Recognize that the central angle is twice the inscribed angle
step 3
Calculate the measure of the central angle BCDBCD using the given inscribed angle ABCABC which is 124124^\circ
step 4
Since BCDBCD is a central angle, BCD=2×ABC=2×124=248BCD = 2 \times ABC = 2 \times 124^\circ = 248^\circ
step 5
Determine the measure of the arc ACDACD that the angle BCDBCD intercepts, which is also 248248^\circ
step 6
Recognize that the angle ADCADC is an inscribed angle that intercepts the same arc ACDACD
step 7
Calculate the measure of the inscribed angle ADCADC which is half of the central angle, so ADC=12×BCD=12×248=124ADC = \frac{1}{2} \times BCD = \frac{1}{2} \times 248^\circ = 124^\circ
The measure of angle ADCADC is 124124^\circ.
Key Concept
Central and Inscribed Angles
The central angle is twice the measure of an inscribed angle that intercepts the same arc. The inscribed angle ADCADC and the central angle BCDBCD intercept the same arc ACDACD, so the measure of ADCADC is half of BCDBCD.

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