Can Two Obtuse Angles Be Complementary to Each Other?

Introduction

In geometry, complementary angles are defined as two angles whose measures add up to 90°. An obtuse angle, on the other hand, is any angle that is greater than 90° but less than 180°. Given these definitions, it becomes clear that there is an inherent conflict when considering whether two obtuse angles can be complementary.

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Definitions

Complementary Angles

• Definition: Two angles are said to be complementary if the sum of their measures equals 90°.
• Example: A 30° angle and a 60° angle are complementary because 30° + 60° = 90°.

Obtuse Angles

• Definition: An angle is considered obtuse if its measure is greater than 90° but less than 180°.
• Example: A 100° angle is obtuse because it exceeds 90°.

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Analyzing the Possibility

Since complementary angles must sum to 90°, let’s consider what happens if we try to pair two obtuse angles:

• By definition, each obtuse angle is greater than 90°.
• If you take the smallest possible obtuse angle (just above 90°), say 91°, even then the sum of two such angles would be 91° + 91° = 182°, which far exceeds 90°.
• Therefore, no matter which obtuse angles you choose, their sum will always be greater than 90°, making it impossible for them to be complementary.

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Conclusion

Based on the definitions and mathematical analysis:

• Answer: No, two obtuse angles cannot be complementary to each other.
• Reason: Complementary angles must add up to 90°, but any obtuse angle is already greater than 90°, so the sum of two obtuse angles will always be greater than 90°.

Understanding these basic concepts helps clarify why the geometric definitions prevent two obtuse angles from ever being complementary.

Disclaimer: This explanation is based on standard Euclidean geometry. For further study, consult a geometry textbook or an academic resource on angle properties.

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