Understanding angles is a fundamental aspect of geometry. Among various types of angles, adjacent angles and complementary angles are two important concepts students frequently encounter. A common question that arises is: Can two adjacent angles be complementary draw figure? This article aims to answer this question in detail, combining definitions, conceptual clarity, mathematical reasoning, and visual illustrations.
Let’s break this down into understandable segments to explore the conditions under which two adjacent angles can be complementary and how to represent this with a proper figure.
What Are Adjacent Angles?
In geometry, adjacent angles are two angles that:
- Share a common vertex,
- Share a common side, and
- Do not overlap—they are next to each other.
They are usually formed when two lines intersect or when a ray is placed inside another angle. For example, if you draw a straight line and a ray originating from its midpoint forming an angle, the two angles on either side of that ray are adjacent.
Example:
If angle AOB and angle BOC are adjacent, they share a common side OB and the same vertex O.
What Are Complementary Angles?
Complementary angles are two angles whose measures add up to exactly 90 degrees.
Mathematically, if ∠X and ∠Y are complementary, then: ∠X + ∠Y = 90°
They don’t necessarily have to be adjacent, but they can be adjacent.
Can Two Adjacent Angles Be Complementary?
Yes, two adjacent angles can be complementary, provided their sum is 90 degrees. When such angles are placed next to each other and add up to 90 degrees, they form a right angle together.
This leads us to the answer: Yes, two adjacent angles can be complementary.
Let’s explore this deeper with visual explanations and examples.
Drawing the Figure
To support the answer with visual aid, let’s draw a simple figure demonstrating two adjacent angles that are complementary.
How to Draw the Figure:
- Draw a horizontal line (say AB).
- Mark a point O anywhere on this line.
- Draw a ray OC from point O going upward, forming an angle with line OB.
- Let ∠COB = 30°
- Draw another ray OD between OC and OB such that ∠AOD = 60°.
Now:
- ∠COB = 30°
- ∠AOD = 60°
- Both share the common vertex O and a common side
- ∠COB and ∠AOD are adjacent angles
- Their sum is 90° → 30° + 60° = 90°
Therefore, the figure illustrates two adjacent and complementary angles.
Real-World Example of Complementary Adjacent Angles
Consider the corner of a rectangular book:
- The corner forms a 90-degree angle.
- If a diagonal line is drawn from the corner, splitting the 90° angle into two parts, say 40° and 50°:
- The two angles are adjacent (they share a side and vertex)
- Their sum is 90°, so they are complementary
This proves that adjacent angles can indeed be complementary in real-world geometries as well.
Mathematical Explanation
Let’s say:
- ∠1 = x°
- ∠2 = 90° – x°
If these two angles are adjacent, they must be side by side and form a right angle when combined.
So:
∠1 + ∠2 = x + (90 – x) = 90°
This satisfies the condition for complementary angles.
If they share a common vertex and side without overlapping, they are also adjacent.
Hence, the pair of angles meet the criteria for both adjacency and complementarity.
Properties of Adjacent Complementary Angles
- They form a right angle when put together.
- They share a vertex and a common arm.
- The non-common arms of the angles form a right angle.
- They appear frequently in geometric constructions involving perpendicular lines.
Difference Between Adjacent, Complementary, and Supplementary Angles
| Type of Angles | Condition | Sum of Angles | Can Be Adjacent? |
|---|---|---|---|
| Adjacent Angles | Share vertex and side | Any | Yes |
| Complementary Angles | Add up to 90° | 90° | Yes |
| Supplementary Angles | Add up to 180° | 180° | Yes |
This table helps to understand the distinctions and overlaps among these types of angles. As seen, adjacent and complementary properties are not mutually exclusive.
Classroom Application
Teachers often use such questions to test multiple concepts at once: angle properties, addition, and diagram drawing.
Sample Problem:
Question: Two adjacent angles are complementary. One angle is 55°. What is the measure of the other angle?
Solution:
- Let the other angle be x.
- x + 55 = 90
- x = 90 – 55 = 35
So the other adjacent angle is 35°.
This approach strengthens mathematical reasoning and geometry skills.
Practice Exercise
Try this:
Draw two adjacent complementary angles where:
- One angle measures 25°
- Find and draw the other
Answer:
- Second angle = 90 – 25 = 65°
- Draw a right angle and divide it with a ray such that one section is 25° and the other is 65°
- You’ve now created a pair of adjacent complementary angles
Importance in Geometry and Real Life
Understanding adjacent and complementary angles is not just academic—it has practical applications:
- Architecture: Right-angle corners are formed using complementary angles.
- Engineering: Machines often need precise angle measurements.
- Interior Design: Furniture and wall alignments depend on correct angular measurements.
Learning to identify and construct adjacent complementary angles equips students with the skills to visualize and apply geometry in real-world settings.
Common Misconceptions
- Complementary angles must be adjacent – False. They don’t have to be, but they can be.
- Adjacent angles are always complementary – False. They can be of any value.
- All right angles are made of two equal complementary angles – Not necessarily; they can be unequal and still be complementary.
Conclusion
Returning to our central question: Can two adjacent angles be complementary draw figure?
The clear and correct answer is: Yes. Two adjacent angles can be complementary if their sum is 90 degrees and they share a common vertex and side without overlapping. This results in a geometric configuration that visually and mathematically forms a right angle.
By understanding the definitions, practicing with diagrams, and exploring real-world applications, students can gain a strong foundation in basic geometric principles. So, next time you see a corner or an L-shaped figure, remember—it could be a perfect example of adjacent complementary angles in action.
