Question

What is the measure of DEF

Answer 1

Answer: OPTION C.

Explanation:

By definition:

`Inscribed\ Angle = (1)/(2) Intercepted\ Arc`

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

`78\°=(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.

Answer 2

The measure of arc DEF is 204° ⇒ answer C

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°