Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.

A right triangle is defined by one angle measuring 90 degrees. The relationships between the sides and angles are governed by the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Understanding these properties is essential for solving right triangles, as they allow for the calculation of unknown side lengths and angles.

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The primary ratios are sine, cosine, and tangent, defined as the ratios of the lengths of the opposite side to the hypotenuse, adjacent side to the hypotenuse, and opposite side to the adjacent side, respectively. These ratios are crucial for finding unknown angles and sides in right triangles.

Angles in triangles can be measured in degrees or radians, with degrees being the more common unit in basic trigonometry. When solving triangles, it is often necessary to convert angles to specific formats, such as minutes (1 degree = 60 minutes), for precision. Understanding how to measure and convert angles is vital for accurately solving problems involving right triangles.