Question

Can you help me understand how to do this correctly?

Answer 1

Let's label the arcs in the circle to be precise.

From the illustration, we can see that the inscribed angle x has intercepted the Arc AB.

Based on the theorem, the measure of the inscribed angles is half the measure of its intercepted arc. Hence, we can say that:

`m\angle x=(1)/(2)m\angle arcAB`

To find the measure of angle x, we need to know first the measure of arc AB.

Notice in the diagram, Arc AB + Arc BC is a semicircle. The total measure of the angles in a semicircle is 180 degrees.

Since we know that Arc BC = 122 degrees, we can subtract this from 180 to get the measure of Arc AB.

`\begin{gathered} \text{ArcAB}+\text{ArcBC}=180 \\ \text{Arc AB}=180-ArcBC \\ \text{Arc AB}=180-122 \\ \text{Arc AB}=58 \end{gathered}`

Hence, the measure of Arc AB is 58 degrees.

Going back to the equation we have formed based on the theorem about inscribed angles, we can plug in the measure of Arc AB and solve for x.

`\begin{gathered} m\angle x=(1)/(2)m\angle ArcAB \\ m\angle x=(1)/(2)(58) \\ m\angle x=29 \end{gathered}`

Hence, the measure of the inscribed angle x is 29 degrees.