1. **Stating the problem:** We have a quadrilateral ABCD with given sides and angles, and we need to find the unknown angle $\theta$ at vertex D. 2. **Known values:** - $\angle BAC = 42^\circ$ - $\angle C = 121^\circ$ - $AB$ is horizontal - $AD = 55$ mm - $DC = 35$ mm - $BC = 60$ mm - $\angle D = \theta$ (unknown) 3. **Approach:** We can use the Law of Cosines and the fact that the sum of interior angles in a quadrilateral is $360^\circ$. 4. **Step 1: Find $\angle B$** Since $AB$ is horizontal and $AC$ forms $42^\circ$ with $AB$, $\angle BAC = 42^\circ$ is given. 5. **Step 2: Use Law of Cosines in triangle BCD to find $\angle D$** Triangle BCD has sides: - $BC = 60$ mm - $CD = 35$ mm - $BD$ unknown We need to find $BD$ first using triangle ABD or ABC. 6. **Step 3: Use Law of Cosines in triangle ABD to find $BD$** Triangle ABD has sides: - $AD = 55$ mm - $AB$ unknown (not given) - $BD$ unknown Since $AB$ is horizontal and no length is given, we cannot directly find $BD$. 7. **Step 4: Use angle sum in quadrilateral** Sum of interior angles in quadrilateral ABCD is: $$\angle A + \angle B + \angle C + \angle D = 360^\circ$$ Given $\angle A = 42^\circ$, $\angle C = 121^\circ$, and $\angle D = \theta$, we need $\angle B$. 8. **Step 5: Find $\angle B$** Since $AB$ is horizontal and $AC$ forms $42^\circ$ with $AB$, $\angle B$ can be found by geometry or given data (not provided). 9. **Conclusion:** Insufficient data to find $\theta$ exactly without length $AB$ or $BD$ or $\angle B$. **Final answer:** Cannot determine $\theta$ with given information.