Question

a right triangle has side lengths ac = 7 inches, bc = 24 inches, and ab = 25 inches. what are the measures of the angles in triangle abc? m∠a≈46.2°, m∠b≈43.8°, m∠c≈90°; m∠a≈74.4°, m∠b≈15.6°, m∠c≈90°; m∠a≈73.0°, m∠b≈17.0°, m∠c≈90°; m∠a≈73.7°, m∠b≈16.3°, m∠c≈90°

Explanation:

Step1: Identify the hypotenuse

The hypotenuse is the longest side. Here, $AB = 25$ inches is the hypotenuse as it is the longest side in right - triangle $ABC$ with $\angle C=90^{\circ}$.

Step2: Use trigonometric ratios to find $\angle A$

We know that $\sin A=\frac{BC}{AB}$. Substituting $BC = 24$ and $AB = 25$, we get $\sin A=\frac{24}{25}=0.96$. Then $A=\sin^{- 1}(0.96)\approx73.7^{\circ}$.

Step3: Use the angle - sum property of a triangle to find $\angle B$

Since the sum of angles in a triangle is $180^{\circ}$ and $\angle C = 90^{\circ}$, $\angle B=180^{\circ}-\angle A - \angle C$. Substituting $\angle A\approx73.7^{\circ}$ and $\angle C = 90^{\circ}$, we get $\angle B=180^{\circ}-73.7^{\circ}-90^{\circ}=16.3^{\circ}$.

Answer:

$m\angle A\approx73.7^{\circ}$, $m\angle B\approx16.3^{\circ}$, $m\angle C\approx90^{\circ}$