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Use the figure to prove that the acute angles of a right triangle are complementary.

To prove that the acute angles of a right triangle are complementary, let's analyze the given right triangle \( \triangle ABC \), where \( \angle C \) is the right angle.

### Proof:
1. **Given**:
   - \( \triangle ABC \) is a right triangle.
   - \( \angle C = 90^\circ \).

2. **To Prove**:
   - The sum of the acute angles \( \angle A \) and \( \angle B \) is \( 90^\circ \). That is, \( \angle A + \angle B = 90^\circ \).

3. **Reasoning**:
   - The sum of the angles in any triangle is always \( 180^\circ \). This is a fundamental property of triangles.

4. **Calculation**:
   - In \( \triangle ABC \), we have:
     \[
     \angle A + \angle B + \angle C = 180^\circ
     \]
   - Since \( \angle C = 90^\circ \), substitute this into the equation:
     \[
     \angle A + \angle B + 90^\circ = 180^\circ
     \]
   - Subtract \( 90^\circ \) from both sides:
     \[
     \angle A + \angle B = 90^\circ
     \]

### Conclusion:
- Therefore, the sum of the acute angles \( \angle A \) and \( \angle B \) in a right triangle is \( 90^\circ \), proving that they are complementary.

Would you like more details or have any questions?

### Related Questions:
1. What is the significance of complementary angles in geometry?
2. How can you prove that the sum of angles in any triangle is \( 180^\circ \)?
3. Can a triangle have more than one right angle? Why or why not?
4. How does the Pythagorean theorem relate to right triangles?
5. What are the properties of isosceles right triangles?
6. How do the angles in an equilateral triangle compare to those in a right triangle?
7. What are some real-world applications of right triangles?
8. How can you use trigonometric ratios to find the angles in a right triangle?

### Tip:
Remember that in any right triangle, the two acute angles always add up to \( 90^\circ \). This concept is fundamental in understanding the properties of right triangles and is widely used in trigonometry.

Learn how to prove that the acute angles of a right triangle are complementary with a detailed mathematical explanation. This problem explores the angle sum property of triangles and its application in right triangles. Suitable for students in Grades 6-8.

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