Complementary angles are two angles whose measures add up to 90 degrees. Given that the two complementary angles are in the ratio [tex]\(2: 3\)[/tex], we will use this information to find the measure of each angle.
1. Set up the problem:
Let the measures of the two angles be [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex].
2. Form an equation:
Since the angles are complementary,
[tex]\[
2x + 3x = 90^\circ
\][/tex]
3. Combine like terms:
[tex]\[
5x = 90^\circ
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find the value of [tex]\(x\)[/tex], divide both sides of the equation by 5:
[tex]\[
x = \frac{90^\circ}{5} = 18^\circ
\][/tex]
5. Find the measures of the angles:
Substitute the value of [tex]\(x\)[/tex] back into the expressions for the angles:
[tex]\[
\text{First angle} = 2x = 2 \times 18^\circ = 36^\circ
\][/tex]
[tex]\[
\text{Second angle} = 3x = 3 \times 18^\circ = 54^\circ
\][/tex]
6. Conclusion:
The two complementary angles are [tex]\(36^\circ\)[/tex] and [tex]\(54^\circ\)[/tex]. They add up to [tex]\(90^\circ\)[/tex], and they are in the ratio [tex]\(2: 3\)[/tex].
