Question

find the measure of arc df. 5x + 10° 70° 11x+2° 50° 90° 100° 140°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, the central angle corresponding to arc $\overset{\frown}{CE}$ and the central angle corresponding to arc $\overset{\frown}{DF}$ are vertical angles. The measure of the central angle corresponding to arc $\overset{\frown}{CE}$ is $70^{\circ}$. Also, we know that the sum of the measures of arcs in a circle is $360^{\circ}$, and the measure of an arc is equal to the measure of its central angle. We can set up an equation using the fact that the sum of the measures of arcs $\overset{\frown}{CE}$, $\overset{\frown}{ED}$, $\overset{\frown}{DF}$, and $\overset{\frown}{FC}$ is $360^{\circ}$. But we can also use the vertical - angle relationship. Since vertical angles are equal, if we consider the central angles of the arcs, the central angle of arc $\overset{\frown}{CE}$ and the central angle of arc $\overset{\frown}{DF}$ are equal.
Let's assume we use the property that vertical angles are equal. The central angle of arc $\overset{\frown}{CE}$ is $70^{\circ}$, and the central angle of arc $\overset{\frown}{DF}$ is also $70^{\circ}$. But if we want to solve it using the arc - length equations:
We know that the measure of an arc is related to the central angle. Let's assume we set up an equation based on the fact that the sum of arcs. However, we can also note that if we consider the relationship between the given arc - angle expressions and the vertical angle.
If we assume that the central angle of arc $\overset{\frown}{CE}$ and arc $\overset{\frown}{DF}$ are vertical angles, we can also solve it in another way.
We know that the measure of an arc is equal to the measure of its central angle.
Let's assume we use the fact that vertical angles are equal. The central angle of arc $\overset{\frown}{CE}$ is $70^{\circ}$, so the measure of arc $\overset{\frown}{DF}$ is also $70^{\circ}$. But if we want to solve it from the perspective of the sum of arcs:
The sum of the measures of arcs in a circle is $360^{\circ}$. Let the measure of arc $\overset{\frown}{CE}=70^{\circ}$, measure of arc $\overset{\frown}{ED} = 5x + 10^{\circ}$, measure of arc $\overset{\frown}{DF}=11x+2^{\circ}$, and measure of arc $\overset{\frown}{FC}$ (not given explicitly but we don't need it if we use vertical - angle property).
Since the central angle of arc $\overset{\frown}{CE}$ and arc $\overset{\frown}{DF}$ are vertical angles, they are equal. So $11x + 2^{\circ}=70^{\circ}$

Step2: Solve for $x$

$11x=70^{\circ}- 2^{\circ}$
$11x = 68^{\circ}$
$x=\frac{68^{\circ}}{11}\approx6.18^{\circ}$
But if we use the vertical - angle property directly, the measure of arc $\overset{\frown}{DF}$ is equal to the measure of the vertical - angle which is $70^{\circ}$
Let's assume we use the fact that the central angle of arc $\overset{\frown}{CE}$ and arc $\overset{\frown}{DF}$ are vertical angles.
The measure of arc $\overset{\frown}{DF}$ is equal to the measure of the central angle that subtends it. Since the central angle of arc $\overset{\frown}{CE}$ and arc $\overset{\frown}{DF}$ are vertical angles and the central angle of arc $\overset{\frown}{CE}$ is $70^{\circ}$, the measure of arc $\overset{\frown}{DF}$ is $70^{\circ}$
However, if we assume that the sum of the non - overlapping arcs around the circle gives us a relationship:
We know that the sum of arcs $\overset{\frown}{CE}+\overset{\frown}{ED}+\overset{\frown}{DF}+\overset{\frown}{FC}=360^{\circ}$
Since vertical angles are equal, the measure of arc $\overset{\frown}{DF}$ is equal to the measure of arc $\overset{\frown}{CE}$ which is $70^{\circ}$
If we consider the fact that the measure of an arc is equal to the measure of its central angle and vertical angles are equal:
The measure of arc $\overset{\frown}{DF}$ is $70^{\circ}$
If we assume that the central angle of arc $\overset{\frown}{CE}$ and arc $\overset{\frown}{DF}$ are vertical angles, we can say that the measure of arc $\overset{\frown}{DF}$ is $70^{\circ}$

Answer:

$70^{\circ}$