Question

MZS = 23

, arc RS = 86

, and RU is tangent to the circle at R. Find mZU.

S

U

Answer 1

`\angle U = 20^\circ`

Explanation:

Given

See attachment

`\angle S = 23`

arc
`RS = 86`

Required

Find
`\angle U`

Let

`O \to` center of the circle

We have that:

arc
`RS = 86`

This implies that:

`\angle SOR = 86^\circ`

and

`\triangle SOR \to` is an isosceles triangle

because:

`SO = OR \to` the radius of the circle

And

`\angle RSO = \angle ORS`

So, we have:

`\angle RSO + \angle ORS + \angle SOR = 180^\circ` --- angles in a triangle

`\angle RSO + \angle ORS + 86^\circ = 180^\circ`

Collect like terms

`\angle RSO + \angle ORS =- 86^\circ + 180^\circ`

`\angle RSO + \angle ORS =94^\circ`

Recall that:
`\angle RSO = \angle ORS`

So:

`\angle RSO + \angle RSO =94^\circ`

`2\angle RSO =94^\circ`

Divide by 2

`\angle RSO =47^\circ`

RU is a tangent.

So:

`\angle ORU = 90^\circ`

Given that:

`\angle S = 23`

Considering
`\triangle SRU`

We have:

`\angle RSU = \angle S = 23^\circ`

So:

`\angle SRU= \angle RSO + \angle ORU`

`\angle SRU= 47 + 90`

`\angle SRU= 137`

Lastly:

`\angle RUS + \angle RSU + \angle SRU =180^(0)`

`\angle RUS + 23 + 137=180^(0)`

Collect like terms

`\angle RUS =180^(0) - 23 - 137`

`\angle RUS =20^(0)`

Hence:

`\angle U = 20^\circ`