Question

70 POINTS!!!!!!!!!! Given: E, F, Q, D∈k(O),O ∈ ED, m∠DFQ = 10°, measure of arc EF = 28° Find: Angles of △EFQ

Answer 1

The measures of angles of triangle EFQ are

1)
`m\angle EFQ=100\°`

2)
`m\angle FEQ=66\°`

3)
`m\angle EQF=14\°`

Explanation:

step 1

Find the measure of arc QD

we know that

The inscribed angle is half that of the arc it comprises.

`m\angle DFQ=(1)/(2)(arc\ QD)`

substitute the given value

`10\°=(1)/(2)(arc\ QD)`

`20\°=(arc\ QD)`

`arc\ QD=20\°`

step 2

Find the measure of arc FQ

we know that

`arc\ QD+arc\ FQ+arc\ EF=180\°` ---> because ED is a diameter (the diameter divide the circle into two equal parts)

substitute the given values

`20\°+arc\ FQ+28\°=180\°`

`arc\ FQ=180\°-48\°=132\°`

step 3

Find the measure of angle EFQ

we know that

The inscribed angle is half that of the arc it comprises.

`m\angle EFQ=(1)/(2)(arc\ QD+arc\ ED)`

substitute the given value

`m\angle EFQ=(1)/(2)(20\°+180\°)=100\°`

step 4

Find the measure of angle FEQ

we know that

The inscribed angle is half that of the arc it comprises.

`m\angle FEQ=(1)/(2)(arc\ FQ)`

substitute the given value

`m\angle FEQ=(1)/(2)(132\°)=66\°`

step 5

Find the measure of angle EQF

we know that

The inscribed angle is half that of the arc it comprises.

`m\angle EQF=(1)/(2)(arc\ EF)`

substitute the given value

`m\angle EQF=(1)/(2)(28\°)=14\°`