Question: 15 38° 35° x Graph description: A quadrilateral-like figure with a diagonal from the top-left vertex to the bottom-right vertex. The left slanted side is labeled 15 and has a tick mark. The top slanted side has a matching tick mark. The angle between the top slanted side and the diagonal at the top-left vertex is 38°. The angle between the bottom side and the diagonal at the bottom-right vertex is 35°. The right side is labeled x. Position hint: center If possible, write an inequality to describe the restrictions on $x$ and which theorem was used to draw this conclusion.

1. **Problem Statement:** We have a quadrilateral with two pairs of congruent sides (marked with tick marks) and angles $38^\circ$ and $35^\circ$ adjacent to the diagonal. We want to find an inequality describing the restrictions on side $x$ using the hinge theorem or its converse. 2. **Relevant Theorem:** The **Hinge Theorem** (also called the SAS Inequality Theorem) states: If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first is larger than the included angle of the second, then the side opposite the larger angle is longer. 3. **Applying the Theorem:** - The two triangles formed by the diagonal have two pairs of congruent sides (both marked with 15 and the top side). - The included angles are $38^\circ$ and $35^\circ$. - Since $38^\circ > 35^\circ$, the side opposite the $38^\circ$ angle (which is $x$) must be longer than the side opposite the $35^\circ$ angle (which is 15). 4. **Inequality:** $$ x > 15 $$ 5. **Conclusion:** By the **Hinge Theorem**, since the included angle opposite $x$ is larger, $x$ must be greater than 15. **Note:** The hinge theorem converse would be used if we knew the side lengths and wanted to compare angles, but here we compare angles to conclude about side lengths. **Final answer:** $$ x > 15 $$