1. **State the problem:** We need to prove that triangles $\triangle GHJ$ and $\triangle GKJ$ are congruent given that $GH \perp HJ$, $GK \perp KJ$, and $GH = GK$. 2. **Identify the right angles:** Since $GH \perp HJ$, angle $\angle GHJ$ is a right angle. Similarly, since $GK \perp KJ$, angle $\angle GKJ$ is a right angle. 3. **Given equal sides:** We know $GH = GK$. 4. **Common side:** Both triangles share side $GJ$. 5. **Apply the Hypotenuse-Leg (HL) theorem:** - In right triangles, if the hypotenuse and one leg are equal, the triangles are congruent. - Here, $GH$ and $GK$ are the hypotenuses of $\triangle GHJ$ and $\triangle GKJ$ respectively. - Side $GJ$ is a leg common to both triangles. 6. **Conclusion:** By HL theorem, $\triangle GHJ \cong \triangle GKJ$. **Final answer:** E. HL