Question

Given: arc IP + arc VK = 86°
Find: ∠KSP

Answer 1

Check the picture below.

using the intersecting chords theorem, let's see, we know that ∡IOP is 43° and since the angle across the junction is a vertical angle, is its twin.

let's recall that a circle has a total of 360°, so if we subtract those two vertical angles, namely 360 - 43 - 43 = 274°.

so the remaining two angles, which are also vertical angles, and therefore twins as well, each one will take 274 ÷ 2 = 137°.

Answer 2

The measure of angle KSP is
`137\°`

Explanation:

step 1

Find the measure of angle ISP

we know that

The measure of the inner angle is the semi-sum of the arcs comprising it and its opposite.

`m<ISP=(1)/(2)(arc\ VK+arc\ IP)`

we have

`(arc\ VK+arc\ IP)=86\°`

substitute

`m<ISP=(1)/(2)(86\°)=43\°`

step 2

Find the measure of angle KSP

we know that

`m<ISP+m<KSP=180\°` ----> by supplementary angles

we have

`m<ISP=43\°`

substitute

`43\°+m<KSP=180\°`

`m<KSP=180\°-43\°=137\°`