In the diagram, $\angle A = 30^\circ$, $\overarc{DE} = 170^\circ$, and $\overarc{BC} = 110^\circ$. Find the measure of arc $CE$, in degrees.
It would help if you proofread your questions before you posted them.
First of all, there is no diagram. You cannot copy and paste here.
Second, you info is hard to understand. What do all the dollar signs ($) signify?
Do you know latex????
To find the measure of arc $CE$, we can follow these steps:
1. Recall that in a circle, an inscribed angle is equal to half the measure of its intercepted arc.
2. Notice that $\angle A$ is an inscribed angle that intercepts arc $\overarc{BC}$. Since $\angle A = 30^\circ$, we know that $\overarc{BC}$ must have a measure of $2 \cdot \angle A = 2 \cdot 30^\circ = 60^\circ$. This means that $\overarc{BC} = 60^\circ$.
3. Now, let's focus on arc $\overarc{DE}$ and arc $\overarc{BC}$. Notice that together, they cover the entire circle. Since the sum of the measures of these two arcs is $360^\circ$, we can set up an equation: $\overarc{DE} + \overarc{BC} = 360^\circ$. Plugging in the given values, we have $170^\circ + 60^\circ = 360^\circ$.
4. Solve the equation for $\overarc{DE}$. Subtract $60^\circ$ from both sides, giving us $\overarc{DE} = 360^\circ - 60^\circ = 300^\circ$.
5. Now, we want to find the measure of arc $CE$. Since arc $\overarc{DE}$ and arc $\overarc{CE}$ are clearly not the same, we should be able to subtract $\overarc{DE}$ from the entire circle to obtain the measure of arc $\overarc{CE}$. Set up an equation: $\overarc{DE} + \overarc{CE} = 360^\circ$.
6. Plug in the value we found for $\overarc{DE}$: $300^\circ + \overarc{CE} = 360^\circ$.
7. Solve for $\overarc{CE}$ by subtracting $300^\circ$ from both sides: $\overarc{CE} = 360^\circ - 300^\circ = 60^\circ$.
Therefore, the measure of arc $CE$ is $\boxed{60^\circ}$.