1. **Problem statement:** We are given a parallelogram KGHJ with angle $\angle K = 38^\circ$. We need to find the size of angle $\angle KGH$. 2. **Recall properties of parallelograms:** - Opposite sides are parallel and equal in length. - Opposite angles are equal. - Adjacent angles are supplementary (sum to $180^\circ$). - Consecutive interior angles between parallel lines are supplementary. 3. **Identify angles:** - Since $KG \parallel HJ$ and $KH \parallel GJ$, $KGH$ is a parallelogram. - Given $\angle K = 38^\circ$, the angle adjacent to $\angle K$ along side $KG$ is $\angle G$. 4. **Calculate $\angle G$:** - Adjacent angles in a parallelogram sum to $180^\circ$. - So, $\angle K + \angle G = 180^\circ$. - Substitute $\angle K = 38^\circ$: $$\angle G = 180^\circ - 38^\circ = 142^\circ$$ 5. **Conclusion:** - The size of angle $\angle KGH$ is $142^\circ$. **Final answer:** $$\boxed{142^\circ}$$