Question

Given: m∠EYL=72°
Find: mEHL , mLVE

Answer 1

Part 1) The measure of arc EHL is
`108\°`

Part 2) The measure of angle LVE is
`54\°`

Explanation:

step 1

Let

x-----> the measure of arc EHL

y----> the measure of arc EVL

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

so

`m<EYL=(1)/(2)(y-x)`

we have

`m<EYL=72\°`

substitute

`72\°=(1)/(2)(y-x)`

`144\°=(y-x)`

`y=144\°+x` ------> equation A

Remember that

`x+y=360\°` -----> equation B ( complete circle)

substitute equation A in equation B and solve for x

`x+(144\°+x)=360\°`

`2x=360\°-144\°`

`x=216\°/2=108\°`

Find the value of y

`y=144\°+x`

`y=144\°+108\°=252\°`

therefore

The measure of arc EHL is
`108\°`

The measure of arc EVL is
`252\°`

step 2

Find the measure of angle LVE

we know that

The inscribed angle measures half that of the arc comprising

Let

x-----> the measure of arc EHL

`m<LVE=(1)/(2)(x)`

we have

`x=108\°`

substitute

`m<LVE=(1)/(2)(108\°)=54\°`