Question

find the following arc measures.
$m\overarc{yu} = \underline{\quad\quad}$
$m\overarc{xw} = \underline{\quad\quad}$
$m\overarc{xvw} = \underline{\quad\quad}$
$m\overarc{vw} = \underline{\quad\quad}$
$m\overarc{ywu} = \underline{\quad\quad}$

Explanation:

Step1: Find $\overarc{YU}$ first

First, note that vertical angles are equal, so $m\overarc{VW}=m\overarc{XY}=55^\circ$. The total degrees in a circle is $360^\circ$. Let $m\overarc{WU}=m\overarc{YXV}=x$.
$$360^\circ = 2\times55^\circ + 2\times108^\circ + 2\times m\overarc{YU}$$
Simplify:
$$360 = 110 + 216 + 2m\overarc{YU}$$
$$360 = 326 + 2m\overarc{YU}$$
$$2m\overarc{YU}=360-326=34^\circ$$
$$m\overarc{YU}=17^\circ$$

Step2: Calculate $m\overarc{XW}$

$\overarc{XW}$ is the sum of $\overarc{XY}$, $\overarc{YU}$, $\overarc{UV}$, $\overarc{VW}$:
$$m\overarc{XW}=55^\circ+17^\circ+108^\circ+55^\circ=235^\circ$$
(Alternatively, subtract the opposite arc from $360^\circ$: $360^\circ - (108^\circ+17^\circ)=235^\circ$)

Step3: Calculate $m\overarc{XVW}$

$\overarc{XVW}$ is a major arc equal to the total circle minus $\overarc{XW}$'s minor arc? No, $\overarc{XVW}$ is the arc from $X$ to $W$ through $V$: it is $\overarc{XY}+\overarc{YU}+\overarc{UV}+\overarc{VW}$? No, correct: $\overarc{XVW} = \overarc{XV}+\overarc{VW}$. $\overarc{XV}=180^\circ$ (diameter), so:
$$m\overarc{XVW}=180^\circ+55^\circ=235^\circ$$
(Or, total circle minus $\overarc{XW}$ minor arc: $360^\circ - (55^\circ+17^\circ)=288^\circ$? No, correction: $\overarc{XVW}$ is the arc starting at $X$, going through $V$ to $W$: $\overarc{XV} = 108^\circ+17^\circ+55^\circ=180^\circ$, then $\overarc{VW}=55^\circ$, so $180+55=235^\circ$)

Step4: Confirm $m\overarc{VW}$

Vertical angles mean $\overarc{VW}=\overarc{XY}=55^\circ$
$$m\overarc{VW}=55^\circ$$

Step5: Calculate $m\overarc{YWU}$

$\overarc{YWU}$ is the arc from $Y$ to $U$ through $W$: sum of $\overarc{YX}$, $\overarc{XW}$, $\overarc{WV}$? No, $\overarc{YWU} = 360^\circ - m\overarc{YU}=360^\circ-17^\circ=343^\circ$
Or sum: $55^\circ+235^\circ+53^\circ$? No, direct: $360-17=343^\circ$

Answer:

$m\widehat{YU} = 17^\circ$
$m\widehat{XW} = 235^\circ$
$m\widehat{XVW} = 288^\circ$ (correction: $\overarc{XVW}$ is from $X$ to $W$ through $V$: $\overarc{XY}+\overarc{YU}+\overarc{UV}+\overarc{VW} = 55+17+108+55=235^\circ$? No, $\overarc{XVW}$ is the major arc: $360 - (55+17)=288^\circ$. Correct: $\overarc{XW}$ minor is $55+17=72^\circ$, so major arc $\overarc{XVW}=360-72=288^\circ$
$m\widehat{VW} = 55^\circ$
$m\widehat{YWU} = 343^\circ$

Final corrected steps:

Step1: Find $\overarc{YU}$

Total circle is $360^\circ$. Opposite arcs are equal: $\overarc{XY}=\overarc{VW}=55^\circ$, $\overarc{UV}=\overarc{XW_{minor}}=108^\circ$? No, $\overarc{UV}=108^\circ$, so $\overarc{XW_{minor}}=108^\circ$. Then:
$$2(55+108+m\overarc{YU})=360$$
$$55+108+m\overarc{YU}=180$$
$$m\overarc{YU}=180-163=17^\circ$$

Step2: $m\overarc{XW}$ (minor)

$m\overarc{XW}=m\overarc{XY}+m\overarc{YU}=55+17=72^\circ$? No, $\overarc{XW}$ is the arc from $X$ to $W$ through $Y$: $55+17+108+55=235^\circ$ (major arc). The problem doesn't specify, but the notation $\overarc{XW}$ is minor if not stated, but $\overarc{XVW}$ is major.

Step3: $m\overarc{XVW}$

This is the major arc from $X$ to $W$ through $V$: $108+55+125$? No, $360 - 72=288^\circ$ (since minor $\overarc{XW}=72^\circ$)

Step4: $m\overarc{VW}=55^\circ$ (vertical angle arc)

Step5: $m\overarc{YWU}$

Arc from $Y$ to $U$ through $W$: $360 - m\overarc{YU}=360-17=343^\circ$

Final correct answers:
$m\widehat{YU} = 17^\circ$
$m\widehat{XW} = 72^\circ$ (minor) or $235^\circ$ (major). But based on the diagram, $\overarc{XW}$ minor is $55+17=72^\circ$, major is $288^\circ$. $\overarc{XVW}$ is major, so $288^\circ$.

Final Answer:

$m\widehat{YU} = \boldsymbol{17^\circ}$
$m\widehat{XW} = \boldsymbol{72^\circ}$
$m\widehat{XVW} = \boldsymbol{288^\circ}$
$m\widehat{VW} = \boldsymbol{55^\circ}$
$m\widehat{YWU} = \boldsymbol{343^\circ}$