Question

what is the measure of DEF

Answer 1

Answer:

The measure of arc DEF is 204° ⇒ answer C

Step-by-step explanation:

* Lets talk about some facts in the circle

- If the vertex of an angle on the circle and the two sides of the

 angle are chords in the circle, then this angle is called  

 an inscribed angle

- Each inscribed angle subtended by the opposite arc, the arc name

 is the starting point and the ending point of the angle

- The measure of any circle is 360°

# Ex: ∠CAB is inscribed angle subtended by arc CB

- There is a relation between the inscribed angle and its  

  subtended arc, the measure of the inscribed angle equals half

  the measure of its subtended arc

* Now lets solve the problem

- ∠DEF is an inscribed angle subtended by arc DF

∴ m∠DEF = (1/2) measure of arc DF

∵ The measure of ∠DEF = 78°

∴ 78° = (1/2) measure of arc DF ⇒ multiply both sides by 2

∴ The measure of arc DF = 78° × 2 = 156°

∵ The measure of arc DF + The measure of arc DEF = The measure of

   the circle

∵ The measure of the circle = 360°

∵ The measure of the arc DF = 156°

∴ 156° + measure of arc DEF = 360° ⇒ subtract 156 from both sides

∴ The measure of arc DEF = 360° - 156° = 204°

* The measure of arc DEF is 204°

Answer 2

Answer: OPTION C.

Step-by-step explanation:

By definition:

$Inscribed\ Angle = \frac{1}{2} Intercepted\ Arc$

Then we can calculate the measure of DF. This is:

$78\°=\frac{1}{2}DF\\\\DF=(2)(78\°)\\\\DF=156\°$

We know that there are 360 degrees in a circle, therefore, in order to find the measure of DEF, we need to make the following subtraction:

$DE$$F$$=360\°-156\°$

$DE$$F$$=204\°$

 You can observe that this matches with the option C.