To solve the problem of finding the measures of two supplementary angles where their measures are in the ratio 11:25, follow these detailed steps:
1. Understand the concept of supplementary angles.
Supplementary angles are two angles whose measures add up to 180 degrees.
2. Set up the ratio.
We are given that the measures of the two angles are in the ratio of 11:25. This means that if the measure of the first angle is represented as 11 parts, then the measure of the second angle is represented as 25 parts.
3. Determine the total parts in the ratio.
Add the parts of the ratio together:
[tex]\[
11 + 25 = 36
\][/tex]
4. Understand that the sum of the supplementary angles is 180 degrees.
Since the two angles are supplementary, their measures add up to 180 degrees.
5. Calculate the measure of each angle based on the ratio.
- To find the measure of the first angle, multiply the total angle sum by the corresponding ratio part:
[tex]\[
\text{First angle} = \left(\frac{11}{36}\right) \times 180
\][/tex]
- Calculate the value:
[tex]\[
\text{First angle} = 55 \, \text{degrees}
\][/tex]
- To find the measure of the second angle, use the ratio part for the second angle:
[tex]\[
\text{Second angle} = \left(\frac{25}{36}\right) \times 180
\][/tex]
- Calculate the value:
[tex]\[
\text{Second angle} = 125 \, \text{degrees}
\][/tex]
6. Conclusion.
The measures of the two supplementary angles are:
[tex]\[
\text{First angle: } 55 \, \text{degrees}
\][/tex]
[tex]\[
\text{Second angle: } 125 \, \text{degrees}
\][/tex]
Thus, the measures of the two supplementary angles in the ratio 11:25 are 55 degrees and 125 degrees.
