Question

Given: measure of arc PIV = 7/2 times measure of arc PKV

PKV

Find: m∠VPJ

Answer 1

Measure of angle 'VPJ' is 140 degrees.

Explanation:

Given:

Measure of arc 'PIV' and measure of arc 'PKV'.

'PIV' = 7/2 times of 'PKV'

Lets say that PKV is 'x'.

⇒
`m(PIV)=(7)/(2) * x`

⇒
`m(PIV)=3.5x`

Note:

A full circle has an arc angle measure of 360.

So,

⇒
`3.5x+x=360`

⇒
`4.5x=360`

⇒
`x=(360)/(4.5)`

⇒
`x=80`

The measure of arc 'PKV' = 80 degrees.

We have to find angle 'VPJ' that is having a linear pair with angle 'VPL'.

So before finding we 'VPJ' have to find 'VPL'.

And

According to the theorem:

The angle formed by the tangent and the chord is half the measure of the intercepted arc.

Then.

⇒
`\angle VPL=(m\ arc\ (PKV))/(2)`

⇒
`\angle VPL=(80)/(2)`

⇒
`\angle VPL=40` degrees

⇒ And from linear pair .

⇒
`m\angle VPL +m \angle VPJ =180`

⇒
`40 +m \angle VPJ =180`

⇒
`m \angle VPJ =180-40`

⇒
`m \angle VPJ =140` degrees.

So measure of angle 'VPJ' is 140 degrees.