Question 1171512
Let's analyze each case using the Law of Sines and the properties of triangles.

**Law of Sines:** p/sin(P) = q/sin(Q) = r/sin(R)

**(a) p = 3, q = 5, angle P = 27 degrees**

1.  **Find sin(Q):**
    * 3/sin(27°) = 5/sin(Q)
    * sin(Q) = (5 * sin(27°)) / 3
    * sin(Q) ≈ (5 * 0.4540) / 3 ≈ 0.7567
    * Q ≈ arcsin(0.7567) ≈ 49.12 degrees

2.  **Check for another possible Q:**
    * Since sin(Q) = sin(180° - Q), there's another possible angle:
    * Q' = 180° - 49.12° ≈ 130.88 degrees

3.  **Check if both Q and Q' are valid:**
    * For Q = 49.12°, P + Q = 27° + 49.12° = 76.12° < 180° (valid)
    * For Q' = 130.88°, P + Q' = 27° + 130.88° = 157.88° < 180° (valid)

4.  **Number of Triangles:**
    * There are **two** non-congruent triangles.

5.  **Size of Angle Q:**
    * Q ≈ 49.12 degrees
    * Q' ≈ 130.88 degrees

**(b) p = 8, q = 5, angle P = 57 degrees**

1.  **Find sin(Q):**
    * 8/sin(57°) = 5/sin(Q)
    * sin(Q) = (5 * sin(57°)) / 8
    * sin(Q) ≈ (5 * 0.8387) / 8 ≈ 0.5242
    * Q ≈ arcsin(0.5242) ≈ 31.61 degrees

2.  **Check for another possible Q:**
    * Q' = 180° - 31.61° ≈ 148.39 degrees

3.  **Check if both Q and Q' are valid:**
    * For Q = 31.61°, P + Q = 57° + 31.61° = 88.61° < 180° (valid)
    * For Q' = 148.39°, P + Q' = 57° + 148.39° = 205.39° > 180° (invalid)

4.  **Number of Triangles:**
    * There is **one** non-congruent triangle.

5.  **Size of Angle Q:**
    * Q ≈ 31.61 degrees

**(c) p = 7, q = 8, angle P = 70 degrees**

1.  **Find sin(Q):**
    * 7/sin(70°) = 8/sin(Q)
    * sin(Q) = (8 * sin(70°)) / 7
    * sin(Q) ≈ (8 * 0.9397) / 7 ≈ 1.0739

2.  **Check for sin(Q) range:**
    * Since sin(Q) > 1, there is no angle Q that satisfies this condition.

3.  **Number of Triangles:**
    * There are **zero** non-congruent triangles.

**(d) p = 10, q = 20, angle P = 30 degrees**

1.  **Find sin(Q):**
    * 10/sin(30°) = 20/sin(Q)
    * sin(Q) = (20 * sin(30°)) / 10
    * sin(Q) = (20 * 0.5) / 10 = 1

2.  **Find angle Q:**
    * Q = arcsin(1) = 90 degrees

3.  **Check for another possible Q:**
    * Q' = 180° - 90° = 90 degrees (same as Q)

4.  **Number of Triangles:**
    * There is **one** non-congruent triangle.

5.  **Size of Angle Q:**
    * Q = 90 degrees