14. The sum of the complement and supplement of a certain angle is two more than twice the said angle, What is the complement of the angle?

Let the angle be denoted as \( x \).

- The **complement** of an angle is \( 90^\circ - x \).
- The **supplement** of an angle is \( 180^\circ - x \).

According to the problem, the sum of the complement and supplement is two more than twice the angle, which gives us the equation:

\[
(90^\circ - x) + (180^\circ - x) = 2x + 2
\]

### Step-by-step solution:

1. Simplify the left-hand side of the equation:
   \[
   90 + 180 - 2x = 2x + 2
   \]
   \[
   270 - 2x = 2x + 2
   \]

2. Move all terms involving \( x \) to one side:
   \[
   270 - 2 = 2x + 2x
   \]
   \[
   268 = 4x
   \]

3. Solve for \( x \):
   \[
   x = \frac{268}{4} = 67^\circ
   \]

### Step 4: Find the complement of the angle
The complement of the angle is:
\[
90^\circ - 67^\circ = 23^\circ
\]

Thus, the complement of the angle is **23 degrees**.

Would you like further clarification or details on this? Here are some related questions you could explore:

1. What is the supplement of an angle if its complement is known?
2. Can an angle have both a complement and a supplement simultaneously?
3. What happens when an angle exceeds \( 90^\circ \) in terms of complement and supplement?
4. How can we visualize the relationship between complement and supplement geometrically?
5. Can this approach be generalized to angles in radians?

**Tip:** Always remember that the sum of complementary angles is \( 90^\circ \), while the sum of supplementary angles is \( 180^\circ \).

The sum of the complement and supplement of a certain angle is two more than twice the said angle. What is the complement of the angle?

Learn how to find the complement of an angle where the sum of the complement and supplement is two more than twice the angle. This problem involves complementary and supplementary angles and basic algebra. Suitable for students in Grades 8-10.

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