Question

what are the missing angle measures in triangle abc?
1.4° and 88.6°
18.6° and 71.4°
35.5° and 54.5°
44.4° and 45.6°
(image of right triangle abc with right angle at c, ac = 7 in, bc = 5 in)

Explanation:

Step1: Identify triangle type

Triangle \( ABC \) is right - angled at \( C \), so \( \angle C = 90^{\circ} \), and we know \( AC = 7\) in, \( BC = 5\) in. We can use trigonometric ratios to find the other angles. Let's find \( \angle B \) first. We know that \( \tan B=\frac{AC}{BC}=\frac{7}{5} \).

Step2: Calculate \( \angle B \)

\( \tan B=\frac{7}{5} = 1.4 \), then \( \angle B=\arctan(1.4)\approx54.5^{\circ} \).

Step3: Calculate \( \angle A \)

Since the sum of angles in a triangle is \( 180^{\circ} \), and \( \angle C = 90^{\circ} \), \( \angle A=180^{\circ}-\angle C - \angle B=180 - 90 - 54.5 = 35.5^{\circ} \). So the missing angles are \( 35.5^{\circ} \) and \( 54.5^{\circ} \).

Answer:

\( 35.5^{\circ} \) and \( 54.5^{\circ} \) (the option with \( 35.5^{\circ} \) and \( 54.5^{\circ} \))