Question

m\widehat{eg}=86{}^{\circ }\angle efg=\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.m\widehat{su}=114{}^{\circ }\angle stu=\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.

Explanation:

Step1: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.
For \(\angle EFG\), the intercepted arc is \(\overset{\frown}{EG}\). Given \(m\overset{\frown}{EG} = 86^{\circ}\).
By the formula \(\angle=\frac{1}{2}\times\) (measure of intercepted arc), we have \(\angle EFG=\frac{1}{2}\times m\overset{\frown}{EG}\).

Step2: Calculate \(\angle EFG\)

Substitute \(m\overset{\frown}{EG} = 86^{\circ}\) into the formula: \(\angle EFG=\frac{1}{2}\times86^{\circ}=43^{\circ}\).

Step3: Use the inscribed - angle theorem for \(\angle STU\)

For \(\angle STU\), the intercepted arc is \(\overset{\frown}{SU}\). Given \(m\overset{\frown}{SU}=114^{\circ}\).
By the formula \(\angle=\frac{1}{2}\times\) (measure of intercepted arc), we have \(\angle STU = \frac{1}{2}\times m\overset{\frown}{SU}\).

Step4: Calculate \(\angle STU\)

Substitute \(m\overset{\frown}{SU}=114^{\circ}\) into the formula: \(\angle STU=\frac{1}{2}\times114^{\circ}=57^{\circ}\).

Answer:

\(\angle EFG = 43^{\circ}\), \(\angle STU=57^{\circ}\)