Find the length of the third side of each right triangle using the Pythagorean theorem.

To find the length of the third side of each right triangle, we use the Pythagorean theorem. This theorem says that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula is:

\[ c^2 = a^2 + b^2 \]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

Let's solve each triangle one by one.

First Triangle:
- Sides: 5 and 12
- Hypotenuse: \( c \)

Using the formula:
\[ c^2 = 5^2 + 12^2 \]
\[ c^2 = 25 + 144 \]
\[ c^2 = 169 \]
\[ c = \sqrt{169} \]
\[ c = 13 \]

Second Triangle:
- Sides: 15 and 20
- Hypotenuse: \( c \)

Using the formula:
\[ c^2 = 15^2 + 20^2 \]
\[ c^2 = 225 + 400 \]
\[ c^2 = 625 \]
\[ c = \sqrt{625} \]
\[ c = 25 \]

Third Triangle:
- Sides: 8 and 15
- Hypotenuse: \( c \)

Using the formula:
\[ c^2 = 8^2 + 15^2 \]
\[ c^2 = 64 + 225 \]
\[ c^2 = 289 \]
\[ c = \sqrt{289} \]
\[ c = 17 \]

Fourth Triangle:
- Sides: 12 and 16
- Hypotenuse: \( c \)

Using the formula:
\[ c^2 = 12^2 + 16^2 \]
\[ c^2 = 144 + 256 \]
\[ c^2 = 400 \]
\[ c = \sqrt{400} \]
\[ c = 20 \]

Final Answer:
The lengths of the third sides are 13, 25, 17, and 20 respectively.