Question

MZS = 23

Answer 1

Answer:

$\angle U = 20^\circ$

Step-by-step 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:

$\angle U = 20^\circ$