1. **State the problem:** We have three angles $d$, $e$, and $f$ formed by three rays originating from a common point. We know one angle is $129^\circ$ and angle $d$ is a right angle ($90^\circ$). We need to find the measures of angles $d$, $e$, and $f$. 2. **Recall the rule:** The sum of angles around a point is always $360^\circ$. 3. **Write the equation:** $$d + e + f + 129^\circ = 360^\circ$$ 4. **Substitute the known value:** $$90^\circ + e + f + 129^\circ = 360^\circ$$ 5. **Simplify the known angles:** $$\cancel{90^\circ} + e + f + \cancel{129^\circ} = 360^\circ$$ $$219^\circ + e + f = 360^\circ$$ 6. **Isolate $e + f$:** $$e + f = 360^\circ - 219^\circ$$ $$e + f = 141^\circ$$ 7. **Use the given angle between $e$ and the vertical ray (129°) to find $e$ and $f$:** Since $e$ and $129^\circ$ are adjacent and $f$ shares a vertex with $e$, and the total around the point is $360^\circ$, the remaining angle $f$ is: $$f = 180^\circ - 129^\circ = 51^\circ$$ 8. **Find $e$:** $$e = 141^\circ - f = 141^\circ - 51^\circ = 90^\circ$$ **Final answers:** $$d = 90^\circ, \quad e = 90^\circ, \quad f = 51^\circ$$