Question: User: 45°
38°
x
15
Graph: a quadrilateral-like shape with a diagonal from the top-left vertex to the bottom-right vertex; the left side and top side have matching tick marks indicating they are congruent, the angle at the top-left is split into 38° and 45°, the bottom side is labeled 15, and 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.
x>38 by the hinge theorem
x>15 by the hinge theorem
x>45 by the hinge theorem
x>15 by the hinge theorem converse
1. **State the problem:**
We have a quadrilateral with two congruent sides and angles of $38^\circ$ and $45^\circ$ adjacent to these sides. The side opposite the $38^\circ$ angle is $15$, and the side opposite the $45^\circ$ angle is $x$. We want to find an inequality describing the restrictions on $x$ using the hinge theorem.
2. **Recall the hinge theorem:**
The hinge theorem states that if two sides of two triangles are congruent, then the triangle with the larger included angle has the longer third side opposite that angle.
3. **Apply the hinge theorem:**
- The two sides adjacent to the angles $38^\circ$ and $45^\circ$ are congruent.
- Since $45^\circ > 38^\circ$, the side opposite $45^\circ$ (which is $x$) must be longer than the side opposite $38^\circ$ (which is $15$).
4. **Write the inequality:**
$$x > 15$$
5. **Conclusion:**
By the hinge theorem, the side $x$ opposite the larger angle $45^\circ$ must be greater than $15$, the side opposite the smaller angle $38^\circ$.
**Final answer:**
$$x > 15 \quad \text{by the hinge theorem}$$