Question

Math

Posted 5 months ago

```
Angle $A$ is circumscribed about circle $O$.
What is the measure of $\angle D$ ?
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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Identify the relationship between the inscribed angle and the central angle that subtends the same arc

step 2

Recognize that angle $A$ is an inscribed angle that subtends arc $BCD$

step 3

Understand that the central angle $O$ that subtends arc $BCD$ is twice the inscribed angle $A$

step 4

Calculate the measure of the central angle $O$ using the given measure of angle $A$: $O = 2 \times A = 2 \times 80^\circ = 160^\circ$

step 5

Note that angle $D$ is also a central angle that subtends arc $BC$

step 6

Realize that the sum of the central angles $O$ and $D$ that subtend the entire circle must be $360^\circ$

step 7

Calculate the measure of angle $D$: $D = 360^\circ - O = 360^\circ - 160^\circ = 200^\circ$

Answer

The measure of $∠D$ is $200^\circ$.

Key Concept

Inscribed Angle and Central Angle

Explanation

The inscribed angle is half the measure of the central angle that subtends the same arc. The sum of all central angles in a circle is $360^\circ$.

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