Answer:
[tex]\angle U = 20^\circ[/tex]
Step-by-step explanation:
Given
See attachment
[tex]\angle S = 23[/tex]
arc [tex]RS = 86[/tex]
Required
Find [tex]\angle U[/tex]
Let
[tex]O \to[/tex] center of the circle
We have that:
arc [tex]RS = 86[/tex]
This implies that:
[tex]\angle SOR = 86^\circ[/tex]
and
[tex]\triangle SOR \to[/tex] is an isosceles triangle
because:
[tex]SO = OR \to[/tex] the radius of the circle
And
[tex]\angle RSO = \angle ORS[/tex]
So, we have:
[tex]\angle RSO + \angle ORS + \angle SOR = 180^\circ[/tex] --- angles in a triangle
[tex]\angle RSO + \angle ORS + 86^\circ = 180^\circ[/tex]
Collect like terms
[tex]\angle RSO + \angle ORS =- 86^\circ + 180^\circ[/tex]
[tex]\angle RSO + \angle ORS =94^\circ[/tex]
Recall that: [tex]\angle RSO = \angle ORS[/tex]
So:
[tex]\angle RSO + \angle RSO =94^\circ[/tex]
[tex]2\angle RSO =94^\circ[/tex]
Divide by 2
[tex]\angle RSO =47^\circ[/tex]
RU is a tangent.
So:
[tex]\angle ORU = 90^\circ[/tex]
Given that:
[tex]\angle S = 23[/tex]
Considering [tex]\triangle SRU[/tex]
We have:
[tex]\angle RSU = \angle S = 23^\circ[/tex]
So:
[tex]\angle SRU= \angle RSO + \angle ORU[/tex]
[tex]\angle SRU= 47 + 90[/tex]
[tex]\angle SRU= 137[/tex]
Lastly:
[tex]\angle RUS + \angle RSU + \angle SRU =180^{0}[/tex]
[tex]\angle RUS + 23 + 137=180^{0}[/tex]
Collect like terms
[tex]\angle RUS =180^{0} - 23 - 137[/tex]
[tex]\angle RUS =20^{0}[/tex]
Hence:
[tex]\angle U = 20^\circ[/tex]
