1. **Problem 1: Which triangles are similar to \(\triangle DEF\)?**\n\nGiven the description, \(\triangle DEF\) is a right triangle inside a larger right triangle \(\triangle DFG\). Points E and F are on the sides of \(\triangle DFG\), and the right angles are at points F and E.\n\n2. **Similarity criteria:** Triangles are similar if their corresponding angles are equal or their sides are proportional. Since \(\triangle DEF\) and \(\triangle DFG\) share angle D and both have right angles, they are similar by AA (Angle-Angle) similarity.\n\n3. **Check each option:**\n- A. \(\triangle DFG\): This is the larger triangle containing \(\triangle DEF\), so they are similar.\n- B. \(\triangle EGF\): Points E, G, F form a triangle but it does not share the same angles as \(\triangle DEF\).\n- C. \(\triangle FEG\): Same as B, not similar.\n- D. \(\triangle GDF\): This is the same as \(\triangle DFG\) but vertices reordered, so similar.\n\n**Answer:** Triangles similar to \(\triangle DEF\) are \(\triangle DFG\) (A) and \(\triangle GDF\) (D).\n\n---\n\n4. **Problem 2: Find the proportion to solve for \(a\) in a right triangle with hypotenuse 18, one leg 16, and other leg \(a\).**\n\n5. **Use Pythagorean theorem:** \(a^2 + 16^2 = 18^2\).\n\n6. **Geometric mean property in right triangles:** The altitude to the hypotenuse creates two segments where the altitude is the geometric mean of the two segments. Alternatively, the leg is the geometric mean of the hypotenuse and the projection of that leg on the hypotenuse.\n\n7. **Check options:**\n- A. \(\frac{34}{a} = \frac{a}{16}\) (34 is not a side, discard)\n- B. \(\frac{a}{2} = \frac{16}{a}\) (2 is not a side, discard)\n- C. \(\frac{a}{18} = \frac{2}{a}\) (2 is not a side, discard)\n- D. \(\frac{18}{a} = \frac{a}{16}\) This matches the geometric mean property: leg \(a\) is the mean proportional between hypotenuse 18 and other leg 16.\n\n**Answer:** D\n\n---\n\n8. **Problem 3: Complete the table for \(\triangle PQR\) with right angle at R and point S on PQ.**\n\n9. **Geometric mean relationships in right triangles:**\n- The altitude from the right angle to the hypotenuse is the geometric mean of the two segments it divides the hypotenuse into.\n- Each leg is the geometric mean of the hypotenuse and the projection of that leg on the hypotenuse.\n\n10. **Given:**\n- Segments PS and QS are parts of PQ (hypotenuse).\n- PR and QR are legs.\n\n11. **Fill in:**\n- PS and QS: altitude from R is geometric mean of PS and QS.\n- PR and QS: PR is geometric mean of PQ and PS.\n- PQ and QS: QR is geometric mean of PQ and QS.\n\n**Final table:**\n| Segments | Geometric Mean |\n|---|---|\n| PS and QS | RS (altitude) |\n| PS and PQ | PR |\n| QS and PQ | QR |