To solve this problem, let's follow a series of logical steps:
1. Understand the concept of supplementary angles:
- Supplementary angles are two angles whose measures add up to 180 degrees.
2. Set up the relationship between the angles:
- Let's denote the measure of the smaller angle as [tex]\( x \)[/tex].
- The measure of the larger angle is given as 3 times the measure of the smaller angle. So, the larger angle can be expressed as [tex]\( 3x \)[/tex].
3. Sum of the angles:
- Since the two angles are supplementary, their sum must equal 180 degrees. Therefore, we can write the equation:
[tex]\[
x + 3x = 180
\][/tex]
4. Simplify the equation:
- Combine like terms on the left side:
[tex]\[
4x = 180
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- To find the value of [tex]\( x \)[/tex], divide both sides of the equation by 4:
[tex]\[
x = \frac{180}{4} = 45
\][/tex]
- Therefore, the measure of the smaller angle is [tex]\( 45 \)[/tex] degrees.
6. Find the measure of the larger angle:
- Since the larger angle is 3 times the measure of the smaller angle, calculate:
[tex]\[
3x = 3(45) = 135
\][/tex]
- Therefore, the measure of the larger angle is [tex]\( 135 \)[/tex] degrees.
In conclusion, the measures of the angles are [tex]\( 45 \)[/tex] degrees and [tex]\( 135 \)[/tex] degrees, respectively.
