1. **Problem Statement:** Prove that triangles $\triangle FGH$ and $\triangle IJH$ are congruent. 2. **Given Information:** - $FG = IJ = 6$ units - $\angle FGH = \angle IJH = 39^\circ$ - Points $F, G, H$ and $I, J, H$ form two triangles sharing vertex $H$. 3. **Step 1: Identify the angles and sides involved.** - $\angle G = \angle J = 39^\circ$ (Given) - $FG = IJ = 6$ (Given) 4. **Step 2: Consider the vertical angles at $H$.** - $\angle GHF$ and $\angle JHI$ are vertical angles formed by intersecting lines $FG$ and $JI$. - Vertical angles are always congruent, so $\angle GHF \cong \angle JHI$. 5. **Step 3: Use the Angle-Side-Angle (ASA) congruence criterion.** - We have two angles and the included side congruent: - $\angle G = \angle J = 39^\circ$ - Side $FG = IJ = 6$ - $\angle GHF = \angle JHI$ (vertical angles) 6. **Step 4: Conclusion** - By ASA, $\triangle FGH \cong \triangle IJH$. **Final answer:** $\boxed{\triangle FGH \cong \triangle IJH}$