To solve this problem, let's follow these steps:
1. Understand complementary angles: Complementary angles are two angles whose measures add up to 90 degrees.
2. The given ratio: The ratio of the angles is 3:7. This means, if one angle is [tex]\(3x\)[/tex], the other angle would be [tex]\(7x\)[/tex].
3. Set up the equation: Since the angles are complementary, their measures add up to 90 degrees.
[tex]\[
3x + 7x = 90
\][/tex]
4. Combine like terms:
[tex]\[
10x = 90
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{90}{10} = 9
\][/tex]
6. Calculate the measures of the angles:
- The smaller angle is [tex]\(3x = 3 \times 9 = 27\)[/tex] degrees.
- The bigger angle is [tex]\(7x = 7 \times 9 = 63\)[/tex] degrees.
7. Supplement of an angle: The supplement of an angle is what, added to the angle, equals 180 degrees. So, find the supplement of the bigger angle (63 degrees):
[tex]\[
180 - 63 = 117
\][/tex]
Thus, the measures of the angles are 27 degrees and 63 degrees, and the supplement of the bigger angle (63 degrees) is 117 degrees.
