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Let's solve the problem:

### Given:
- Right triangle \( \triangle ABC \) where:
  - \( AC = 5 \) (opposite side),
  - \( BC = 12 \) (adjacent side),
  - \( \angle C = 90^\circ \) (right angle).

We need to:
1. Determine the measures of angles \( A \) and \( B \).
2. Find the hypotenuse \( AB \).

---

### Step 1: Find the length of side \( AB \) (hypotenuse)
Using the **Pythagorean theorem**:

\[
AB^2 = AC^2 + BC^2
\]

Substitute the given values:

\[
AB^2 = 5^2 + 12^2 = 25 + 144 = 169
\]

\[
AB = \sqrt{169} = 13
\]

Thus, \( AB = 13 \).

---

### Step 2: Calculate the angles \( A \) and \( B \)
We use trigonometric ratios. 

#### Finding \( \angle A \):
Using the **tangent ratio**:

\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{BC} = \frac{5}{12}
\]

Find \( A \) using the inverse tangent:

\[
A = \tan^{-1}\left(\frac{5}{12}\right)
\]

Using a calculator:

\[
A \approx 22.62^\circ
\]

#### Finding \( \angle B \):
Since the angles in a triangle sum to \( 180^\circ \), and \( \angle C = 90^\circ \):

\[
B = 90^\circ - A = 90^\circ - 22.62^\circ \approx 67.38^\circ
\]

---

### Final Answers:
1. **Angles:**
   - \( \angle A \approx 22.62^\circ \),
   - \( \angle B \approx 67.38^\circ \),
   - \( \angle C = 90^\circ \) (given).

2. **Length of \( AB \):**
   - \( AB = 13 \).

---

Would you like a detailed breakdown of the trigonometric steps or further clarifications?

---

### Related Questions:
1. How is the tangent ratio derived, and what are its applications?
2. What is the significance of the Pythagorean theorem in solving right triangles?
3. How would the solution change if one side were unknown?
4. Can these steps apply to non-right triangles? Why or why not?
5. What other trigonometric ratios could be used for solving this triangle?

---

### Tip:
Always check if a triangle is a common Pythagorean triple (like 5-12-13) to simplify calculations.

Refer to the figure below.
1. What are the measures of the missing angles? Angles A, B, and C.
2. What is the length of Side AB?

Learn how to solve a right triangle with given side lengths AC = 5 and BC = 12. Calculate the hypotenuse (AB = 13) using the Pythagorean theorem and find angles A (~22.62°) and B (~67.38°) using trigonometric ratios. Perfect for Grades 8-10.

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