Final answer:
To list triangles' angles and sides from smallest to largest, use the Pythagorean theorem and trigonometric ratios. The smallest angle is opposite the shortest side. After finding angles with trigonometric functions, order them with sides from smallest to largest.
Step-by-step explanation:
To list the angles and sides of each triangle in order from smallest to largest, you should first understand the properties of triangles. For any triangle, the sum of internal angles is always 180 degrees. In a right triangle, like the ones mentioned, the sides relate to each other based on the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
According to trigonometric relationships, the size of an angle in a right triangle is directly related to the lengths of the sides. The smallest angle is opposite the shortest side, and the largest angle, which is always 90 degrees in a right triangle, is opposite the longest side, the hypotenuse. With this understanding, if you have the lengths of two sides, you can use trigonometric ratios such as sine (sin), cosine (cos), and tangent (tan) to determine the angles.
For example, if you know the adjacent side (x) and the opposite side (y) to the angle you are trying to find, you can use tan to find the angle, where tan of the angle equals y/x. Similarly, sin of the angle equals y/hypotenuse, and cos of the angle equals x/hypotenuse.
By using these ratios, after finding the angles, list them together with the sides starting from the smallest to the largest as follows: smallest side, next largest side, hypotenuse, smallest angle, and largest angle (which is the right angle).