`angleX` and `angleY` are supplementary angles. If `angleX:angleY=25:11`, then find `angleX` and `angleY`.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understand the relationship between the angles**:
Since angle X and angle Y are supplementary angles, we know that:
\[ \text{angle X} + \text{angle Y} = 180^\circ \]

2. **Set up the ratio**:
We are given that the ratio of angle X to angle Y is:
\[ \frac{\text{angle X}}{\text{angle Y}} = \frac{25}{11} \]
This can be rewritten as:
\[ \text{angle X} = \frac{25}{11} \times \text{angle Y} \]

3. **Substitute angle X in the supplementary angle equation**:
Now, we can substitute the expression for angle X into the supplementary angle equation:
\[ \frac{25}{11} \times \text{angle Y} + \text{angle Y} = 180^\circ \]

4. **Combine the terms**:
To combine the terms on the left side, we can factor out angle Y:
\[ \text{angle Y} \left(\frac{25}{11} + 1\right) = 180^\circ \]
Converting 1 into a fraction with a common denominator:
\[ 1 = \frac{11}{11} \]
So we have:
\[ \text{angle Y} \left(\frac{25}{11} + \frac{11}{11}\right) = 180^\circ \]
This simplifies to:
\[ \text{angle Y} \left(\frac{36}{11}\right) = 180^\circ \]

5. **Solve for angle Y**:
To isolate angle Y, multiply both sides by the reciprocal of \(\frac{36}{11}\):
\[ \text{angle Y} = 180^\circ \times \frac{11}{36} \]
Simplifying this:
\[ \text{angle Y} = \frac{1980}{36} = 55^\circ \]

6. **Find angle X**:
Now that we have angle Y, we can find angle X using the supplementary angle equation:
\[ \text{angle X} = 180^\circ - \text{angle Y} = 180^\circ - 55^\circ = 125^\circ \]

### Final Answer:
- \(\text{angle X} = 125^\circ\)
- \(\text{angle Y} = 55^\circ\)