In a right-angled isosceles triangle, the measures of the angles are _______.

Explanation

To find the measures of the angles in a right-angled isosceles triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of the Triangle**:
A right-angled isosceles triangle has one angle that is a right angle (90 degrees) and the other two angles are equal.

**Hint**: Remember that in an isosceles triangle, two angles are always the same.

2. **Identify the Right Angle**:
In this triangle, we know that one angle is 90 degrees. Let's denote the angles as A, B, and C. We can say that angle B = 90 degrees.

**Hint**: A right-angled triangle has one angle that measures exactly 90 degrees.

3. **Use the Angle Sum Property**:
The sum of all angles in any triangle is 180 degrees. Therefore, we can write the equation:
\[ A + B + C = 180 \]
Substituting the known value of angle B:
\[ A + 90 + C = 180 \]

**Hint**: The total measure of angles in any triangle is always 180 degrees.

4. **Simplify the Equation**:
Rearranging the equation gives us:
\[ A + C = 180 - 90 \]
\[ A + C = 90 \]

**Hint**: After finding one angle, subtract it from 180 to find the sum of the other two angles.

5. **Determine the Equal Angles**:
Since angles A and C are equal (because it is an isosceles triangle), we can say:
\[ A = C \]
Therefore, we can write:
\[ A + A = 90 \]
or
\[ 2A = 90 \]

**Hint**: If two angles are equal, you can combine them into one expression.

6. **Solve for Angle A**:
Dividing both sides by 2 gives:
\[ A = \frac{90}{2} = 45 \]

**Hint**: To find the measure of one angle, divide the total by the number of equal angles.

7. **Find Angle C**:
Since angle C is equal to angle A, we have:
\[ C = 45 \]

**Hint**: In an isosceles triangle, the two equal angles will have the same measure.

8. **Conclusion**:
The measures of the angles in a right-angled isosceles triangle are:
- Angle A = 45 degrees
- Angle B = 90 degrees
- Angle C = 45 degrees

Therefore, the final answer is:
**The measures of the angles are 45 degrees, 90 degrees, and 45 degrees.**