Introduction
In geometry, complementary angles are defined as two angles whose measures add up to 90°. When two angles share a common vertex and side, they are said to be adjacent. This raises the question: Can two adjacent angles be complementary? The answer is yes—as long as the measures of the two angles add up to 90°. In this article, we’ll explain the concept and provide a simple diagram to illustrate how two adjacent angles can be complementary.
Understanding Complementary and Adjacent Angles
Complementary Angles
- Definition: Two angles are complementary if the sum of their measures is exactly 90°.
- Example: If one angle measures 40°, then its complementary angle must measure 50° (because 40° + 50° = 90°).
Adjacent Angles
- Definition: Two angles are adjacent if they share a common vertex and one common side, and do not overlap.
- Example: If you have a right angle (90°) that is split into two parts by a ray, the two resulting adjacent angles are complementary if their sum is 90°.
How Two Adjacent Angles Can Be Complementary
Imagine a right angle, which by definition measures 90°. If a ray is drawn inside the right angle, it splits the angle into two adjacent angles. Let’s denote these angles as α and β. For them to be complementary, the following must hold:
\alpha + \beta = 90^\circ
For example, if α = 35°, then β must equal 55°.
Diagram Illustrating Complementary Adjacent Angles
Below is a simple diagram showing a right angle (90°) split into two adjacent angles, α and β, which add up to 90°:
O
/|\
/ | \
/ | \
α / | \ β
/ | \
/_____|_____\
A C B
- O is the vertex of the right angle.
- OA and OB are the two sides forming the 90° angle.
- OC is the ray that divides the right angle into two adjacent angles: ∠AOC (α) and ∠COB (β).
- Since the entire angle ∠AOB is 90°, it follows that:
\alpha + \beta = 90^\circ
This diagram visually confirms that two adjacent angles can indeed be complementary.
Conclusion
Yes, two adjacent angles can be complementary if the sum of their measures is exactly 90°. The diagram above illustrates a classic example: a right angle split by an interior ray into two adjacent angles, α and β, which satisfy the equation α + β = 90°. Understanding this fundamental concept helps in various areas of geometry and aids in solving problems related to angles and their relationships.
Disclaimer: This article is for educational purposes and is based on standard Euclidean geometry.
Also Check:
• Can Two Obtuse Angles Be Complementary to Each Other?
• Can A Triangle Have Two Right Angles? Exploring the Limits of Triangle Geometry
• How Many Altitudes Can a Triangle Have? An Easy-to-Understand Guide
• What Can You Say About the Motion of an Object? An In-Depth Exploration
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