besides the 90° angle measure, what are the other two angle measures of a right triangle with side lengths…

Explanation:

Step1: Use trigonometric ratios

In a right - triangle, if the sides are (a = 5), (b=12), and (c = 13) (hypotenuse). Let (\theta_1) be the angle opposite to side (a). Then (\sin\theta_1=\frac{a}{c}), and (\theta_2) be the angle opposite to side (b), (\sin\theta_2=\frac{b}{c}).

Step2: Calculate (\theta_1)

We know that (\sin\theta_1=\frac{5}{13}). Using the inverse - sine function (\theta_1=\sin^{- 1}(\frac{5}{13})). [ \theta_1=\sin^{-1}\left(\frac{5}{13}\right)\approx23^{\circ} ]

Step3: Calculate (\theta_2)

Since the sum of angles in a triangle is (180^{\circ}) and one angle is (90^{\circ}). Also, using (\sin\theta_2=\frac{12}{13}), (\theta_2=\sin^{-1}(\frac{12}{13})) [ \theta_2=\sin^{-1}\left(\frac{12}{13}\right)\approx67^{\circ} ] Another way: Since (\theta_1+\theta_2 = 90^{\circ}) (complementary angles in a right - triangle), and if (\theta_1\approx23^{\circ}), then (\theta_2=90^{\circ}-\theta_1\approx90 - 23=67^{\circ})

Answer:

23° and 67°