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A right triangle has side lengths [tex]$AC = 7$[/tex] inches, [tex]$BC = 24$[/tex] inches, and [tex][tex]$AB = 25$[/tex][/tex] inches. What are the measures of the angles in triangle [tex]$ABC$[/tex]?

A. [tex]m \angle A \approx 46.2^{\circ}, m \angle B \approx 43.8^{\circ}, m \angle C \approx 90^{\circ}[/tex]

B. [tex]m \angle A \approx 73.0^{\circ}, m \angle B \approx 17.0^{\circ}, m \angle C \approx 90^{\circ}[/tex]

C. [tex]m \angle A \approx 73.7^{\circ}, m \angle B \approx 16.3^{\circ}, m \angle C \approx 90^{\circ}[/tex]

D. [tex]m \angle A \approx 74.4^{\circ}, m \angle B \approx 15.6^{\circ}, m \angle C \approx 90^{\circ}[/tex]



Answer :

GinnyAnswer
To solve the problem of finding the measures of the angles in triangle [tex]\(ABC\)[/tex] where [tex]\(AC = 7\)[/tex] inches, [tex]\(BC = 24\)[/tex] inches, and [tex]\(AB = 25\)[/tex] inches, follow these steps:

### Step 1: Identify the Given Information
We are given:
- [tex]\(AC = 7\)[/tex] inches (one leg of the triangle)
- [tex]\(BC = 24\)[/tex] inches (the other leg of the triangle)
- [tex]\(AB = 25\)[/tex] inches (the hypotenuse of the triangle)

Since the triangle is a right triangle, we have:
- [tex]\(\angle C = 90^\circ\)[/tex]

### Step 2: Use Trigonometric Ratios to Find Angles
1. Finding [tex]\(\angle A\)[/tex]:
[tex]\[ \sin(\angle A) = \frac{AC}{AB} = \frac{7}{25} \][/tex]
[tex]\[ \angle A = \arcsin\left(\frac{7}{25}\right) \approx 16.3^\circ \][/tex]

2. Finding [tex]\(\angle B\)[/tex]:
Using the complementary angle relationship in a right triangle:
[tex]\[ \angle B = 90^\circ - \angle A = 90^\circ - 16.3^\circ \approx 73.7^\circ \][/tex]

### Step 3: Conclude the Measures of the Angles
From the calculations:
- [tex]\(\angle A \approx 16.3^\circ\)[/tex]
- [tex]\(\angle B \approx 73.7^\circ\)[/tex]
- [tex]\(\angle C = 90^\circ\)[/tex] (since it is a right triangle)

Thus, the correct measures of the angles in triangle [tex]\(ABC\)[/tex] are:
- [tex]\(m \angle A \approx 16.3^\circ\)[/tex]
- [tex]\(m \angle B \approx 73.7^\circ\)[/tex]
- [tex]\(m \angle C = 90^\circ\)[/tex]

### Answer
The correct option is:
[tex]\[ \boxed{m \angle A \approx 73.7^\circ, m_{\angle} B \approx 16.3^\circ, m \angle C \approx 90^\circ} \][/tex]

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