1. **Problem Statement:** Find the measures of angles 1 through 22 in the circle with center G, diameter AD, and given arc measures $mBC=87^\circ$, $mFE=42^\circ$, $mED=48^\circ$, and $mCD=45^\circ$. 2. **Key Formulas and Rules:** - Central angle measure equals the measure of its intercepted arc. - Inscribed angle measure is half the measure of its intercepted arc. - Diameter creates a $180^\circ$ arc. - Angles formed by intersecting chords can be found using arc sums. 3. **Calculate missing arcs:** - Since $AD$ is diameter, arc $AD=180^\circ$. - Total circle $=360^\circ$. - Sum known arcs: $mBC + mFE + mED + mCD = 87 + 42 + 48 + 45 = 222^\circ$. - Remaining arcs $= 360 - 222 = 138^\circ$ (includes arcs $AB$ and $FA$). 4. **Find each angle:** - $m\angle1$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle1=\frac{42}{2}=21^\circ$. - $m\angle2$: Inscribed angle intercepting arc $ED=48^\circ$, so $m\angle2=24^\circ$. - $m\angle3$: Inscribed angle intercepting arc $CD=45^\circ$, so $m\angle3=22.5^\circ$. - $m\angle4$: Central angle intercepting arc $AD=180^\circ$, so $m\angle4=180^\circ$. - $m\angle5$: Inscribed angle intercepting arc $BC=87^\circ$, so $m\angle5=43.5^\circ$. - $m\angle6$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle6=21^\circ$. - $m\angle7$: Inscribed angle intercepting arc $ED=48^\circ$, so $m\angle7=24^\circ$. - $m\angle8$: Inscribed angle intercepting arc $CD=45^\circ$, so $m\angle8=22.5^\circ$. - $m\angle9$: Central angle intercepting arc $BC=87^\circ$, so $m\angle9=87^\circ$. - $m\angle10$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle10=21^\circ$. - $m\angle11$: Inscribed angle intercepting arc $ED=48^\circ$, so $m\angle11=24^\circ$. - $m\angle12$: Inscribed angle intercepting arc $CD=45^\circ$, so $m\angle12=22.5^\circ$. - $m\angle13$: Inscribed angle intercepting arc $BC=87^\circ$, so $m\angle13=43.5^\circ$. - $m\angle14$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle14=21^\circ$. - $m\angle15$: Inscribed angle intercepting arc $ED=48^\circ$, so $m\angle15=24^\circ$. - $m\angle16$: Inscribed angle intercepting arc $CD=45^\circ$, so $m\angle16=22.5^\circ$. - $m\angle17$: Central angle intercepting arc $BC=87^\circ$, so $m\angle17=87^\circ$. - $m\angle18$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle18=21^\circ$. - $m\angle19$: Inscribed angle intercepting arc $ED=48^\circ$, so $m\angle19=24^\circ$. - $m\angle20$: Inscribed angle intercepting arc $CD=45^\circ$, so $m\angle20=22.5^\circ$. - $m\angle21$: Central angle intercepting arc $BC=87^\circ$, so $m\angle21=87^\circ$. - $m\angle22$: Inscribed angle intercepting arc $FE=42^\circ$, so $m\angle22=21^\circ$. **Final answers:** $$\begin{aligned} m\angle1 &= 21^\circ, & m\angle2 &= 24^\circ, & m\angle3 &= 22.5^\circ, & m\angle4 &= 180^\circ, \\ m\angle5 &= 43.5^\circ, & m\angle6 &= 21^\circ, & m\angle7 &= 24^\circ, & m\angle8 &= 22.5^\circ, \\ m\angle9 &= 87^\circ, & m\angle10 &= 21^\circ, & m\angle11 &= 24^\circ, & m\angle12 &= 22.5^\circ, \\ m\angle13 &= 43.5^\circ, & m\angle14 &= 21^\circ, & m\angle15 &= 24^\circ, & m\angle16 &= 22.5^\circ, \\ m\angle17 &= 87^\circ, & m\angle18 &= 21^\circ, & m\angle19 &= 24^\circ, & m\angle20 &= 22.5^\circ, \\ m\angle21 &= 87^\circ, & m\angle22 &= 21^\circ. \end{aligned}$$