To solve the problem where the sum of an angle and the complementary angle of [tex]\(15^\circ\)[/tex] is [tex]\(100^\circ\)[/tex], let's break it down step-by-step:
1. Understanding complementary angles:
- Complementary angles are two angles whose measures add up to [tex]\(90^\circ\)[/tex].
- Therefore, if one of the angles is [tex]\(15^\circ\)[/tex], the complementary angle would be [tex]\(90^\circ - 15^\circ\)[/tex].
2. Setting up the problem:
- Let [tex]\(x\)[/tex] be the unknown angle.
- The complement of angle [tex]\(x\)[/tex] would be [tex]\(90^\circ - x\)[/tex] because they must add up to [tex]\(90^\circ\)[/tex] (as they are complementary).
- According to the problem, the sum of angle [tex]\(x\)[/tex] and the complementary angle of [tex]\(15^\circ\)[/tex] is [tex]\(100^\circ\)[/tex].
3. Formulate the equation:
- The given complementary angle is [tex]\(15^\circ\)[/tex].
- So, we need to find the angle [tex]\(x\)[/tex] such that the sum of [tex]\(x\)[/tex] and [tex]\(90^\circ - x\)[/tex] plus additional [tex]\(15^\circ\)[/tex] equals [tex]\(100^\circ\)[/tex].
4. Solving the equation:
- Combine the given information into an equation:
[tex]\[
x + (90^\circ - x) + 15^\circ = 100^\circ
\][/tex]
- Simplify the equation by combining like terms:
[tex]\[
90^\circ + 15^\circ = 100^\circ
\][/tex]
- Therefore, [tex]\(x + 15^\circ = 85^\circ\)[/tex]
[tex]\[
x + 15^\circ = 85^\circ
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 85^\circ
\][/tex]
5. Determine the complementary angle:
- If [tex]\(x = 85^\circ\)[/tex], then the complementary angle is:
[tex]\[
90^\circ - x = 90^\circ - 85^\circ = 5^\circ
\][/tex]
Thus, the angle [tex]\(x\)[/tex] is [tex]\(85^\circ\)[/tex], and its complementary angle is [tex]\(5^\circ\)[/tex].
