Question

Math

Posted 2 months ago

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A circle is centered on point $B$. Points $A, C$ and $D$ lie on its circumference.
If $\angle A B C$ measures $124^{\circ}$, what does $\angle A D C$ measure?
$\square$
```

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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 $BCD$ using the given inscribed angle $ABC$ which is $124^\circ$

step 4

Since $BCD$ is a central angle, $BCD = 2 \times ABC = 2 \times 124^\circ = 248^\circ$

step 5

Determine the measure of the arc $ACD$ that the angle $BCD$ intercepts, which is also $248^\circ$

step 6

Recognize that the angle $ADC$ is an inscribed angle that intercepts the same arc $ACD$

step 7

Calculate the measure of the inscribed angle $ADC$ which is half of the central angle, so $ADC = \frac{1}{2} \times BCD = \frac{1}{2} \times 248^\circ = 124^\circ$

Answer

The measure of angle $ADC$ is $124^\circ$.

Key Concept

Central and Inscribed Angles

Explanation

The central angle is twice the measure of an inscribed angle that intercepts the same arc. The inscribed angle $ADC$ and the central angle $BCD$ intercept the same arc $ACD$, so the measure of $ADC$ is half of $BCD$.

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