The problem gives the measures of three central angles in a circle: $m\angle QNS = 80^\circ$, $m\angle SNU = 65^\circ$, and $m\angle QNW = 95^\circ$. We need to find the measures of the following arcs: $m\widehat{SW}$, $m\widehat{QUW}$, and $m\widehat{UW}$.

1. Problem Description

The problem gives the measures of three central angles in a circle: $m\angle QNS = 80^\circ$, $m\angle SNU = 65^\circ$, and $m\angle QNW = 95^\circ$. We need to find the measures of the following arcs: $m\widehat{SW}$, $m\widehat{QUW}$, and $m\widehat{UW}$.

2. Solution Steps

a. To find $m\widehat{SW}$, we first need to find $m\angle SNW$. Since $m\angle QNW = 95^\circ$ and $m\angle QNS = 80^\circ$, we can find $m\angle SNW$ by subtraction.

$m\angle SNW = m\angle QNW - m\angle QNS$

$m\angle SNW = 95^\circ - 80^\circ = 15^\circ$

Now we can find $m\widehat{SW}$. The measure of an arc is equal to the measure of its central angle.

Therefore, $m\widehat{SW} = m\angle SNW = 15^\circ$.

b. To find $m\widehat{QUW}$, we use the property that the measure of a central angle equals the measure of its intercepted arc. $m\widehat{QW} = m\angle QNW = 95^\circ$.

We are given that $m\angle QNS = 80^\circ$ and $m\angle SNU = 65^\circ$. Thus $m\widehat{QS} = 80^\circ$ and $m\widehat{SU} = 65^\circ$.

To find $m\widehat{QU}$, we have $m\widehat{QU}=m\widehat{QW} - m\widehat{UW}$.

We know $m\angle QNW = 95^\circ$. To find $m\widehat{UW}$, we need to find $m\angle UNW$. The sum of the central angles around point N is $360^\circ$. Therefore:

$m\angle QNS + m\angle SNU + m\angle UNW + m\angle QNW = 360^\circ$

$80^\circ + 65^\circ + m\angle UNW + 95^\circ = 360^\circ$

$240^\circ + m\angle UNW = 360^\circ$

$m\angle UNW = 360^\circ - 240^\circ = 120^\circ$.

So, $m\widehat{UW} = 120^\circ$.

Then $m\widehat{QU} = m\widehat{QW} + m\widehat{UW} = 95^\circ$. Therefore $m\widehat{QUW} = m\widehat{QU}+m\widehat{UW} = 95^\circ + 120^\circ=215^\circ$.

c. We already found $m\widehat{UW}$ in part b, which is $120^\circ$.

Therefore, $m\widehat{UW} = 120^\circ$.

3. Final Answer

a. $m\widehat{SW} = 15^\circ$

b. $m\widehat{QUW} = 215^\circ$

c. $m\widehat{UW} = 120^\circ$