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1. a = 5,\ b = 12,\ c = 13

   Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓

2. P = a + b + c = 30

   Perimeter.

3. s = \tfrac{P}{2} = 15,\quad A = \sqrt{s(s-a)(s-b)(s-c)} = 30

   Heron's formula for the area.

4. \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{12}{13} \Rightarrow A \approx 22.620^\circ

   Law of cosines for angle A (opposite side a).

5. \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{5}{13} \Rightarrow B \approx 67.380^\circ

   Law of cosines for angle B (opposite side b).

6. \cos C = \frac{a^2 + b^2 - c^2}{2ab} = 0 \Rightarrow C \approx 90.000^\circ

   Law of cosines for angle C (opposite side c).

7. A + B + C = 180.0^\circ

   The angles add to 180° — a right, scalene triangle.