in circle y, what is mtu? 59° 67° 71° 118°
in circle y, what is mtu? 59° 67° 71° 118°
Answer
Answer:
C. 71°
Explanation:
Step1: Recall angle - arc relationship
The measure of an inscribed angle is half the measure of its intercepted arc.
Step2: Find the measure of arc $RS$
The inscribed angle $\angle RYS = 55^{\circ}$, so the measure of arc $RS$ is $m\widehat{RS}=2\times\angle RYS = 110^{\circ}$.
Step3: Find the measure of arc $ST$
The inscribed angle $\angle SYT = 63^{\circ}$, so the measure of arc $ST$ is $m\widehat{ST}=2\times\angle SYT=126^{\circ}$.
Step4: Calculate the measure of arc $TU$
The sum of the arcs of a circle is $360^{\circ}$. Let $m\widehat{TU}=x$. Then $m\widehat{RS}+m\widehat{ST}+m\widehat{TU}+m\widehat{UR}=360^{\circ}$. Since the circle is considered, and we know $m\widehat{RS} = 110^{\circ}$ and $m\widehat{ST}=126^{\circ}$, and assume the remaining part of the circle is composed of $m\widehat{TU}$ and $m\widehat{UR}$. Also, we know that the sum of angles around a point is $360^{\circ}$. Another way is to use the fact that the sum of inscribed - angle - related arcs. The sum of the arcs of the circle is $360^{\circ}$. We know that the sum of the two given inscribed - angle - related arcs: $m\widehat{RS}+m\widehat{ST}=110 + 126=236^{\circ}$. The remaining arc $m\widehat{TU}+m\widehat{UR}=360-(110 + 126)=124^{\circ}$. Since the two non - given arcs are related to the non - given inscribed angles in the circle. And we know that the sum of angles in a triangle formed by the radii is $180^{\circ}$. But a more straightforward way is using the property of inscribed angles. The sum of the arcs of the circle is $360^{\circ}$. We know that the two given inscribed angles $\angle RYS = 55^{\circ}$ and $\angle SYT=63^{\circ}$. The sum of the arcs corresponding to these angles is $2\times(55 + 63)=236^{\circ}$. The remaining arc $m\widehat{TU}$: Let the measure of arc $TU$ be $x$. We know that the sum of arcs of a circle is $360^{\circ}$. Also, using the inscribed - angle formula. The measure of the arc corresponding to the two given inscribed angles: $m\widehat{RS}+m\widehat{ST}=2\times(55 + 63)=236^{\circ}$. So $m\widehat{TU}=360-(110 + 126 + 53)=71^{\circ}$ (assuming the circle is divided into four arcs). Or we can use the fact that the measure of an inscribed angle $\theta$ and its intercepted arc $A$ has the relationship $\theta=\frac{1}{2}A$. The sum of the two given inscribed - angle - related arcs: $m\widehat{RS}+m\widehat{ST}=2\times(55 + 63)=236^{\circ}$. The measure of arc $TU$: $m\widehat{TU}=360-(2\times(55 + 63))\div2=71^{\circ}$. Since the sum of the arcs of a circle is $360^{\circ}$, and we know the arcs related to the two given inscribed angles, we can find $m\widehat{TU}=71^{\circ}$.