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Prove that a triangle must have atleast two acute angles.?
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Prove that a triangle must have atleast two acute angles.?
Without two acute angles a triangle cannot be formed as acute angle is less than 90 degree. for ex . 100, 30, 50 , It contains two acute angle. It can also contains 3 acute angles for ex - 70, 30, 80 , ......... upvote if this helps.
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Prove that a triangle must have atleast two acute angles.?
Introduction
A triangle is a polygon with three sides and three angles. In order to prove that a triangle must have at least two acute angles, we will use the properties and definitions of triangles.
Definition of an acute angle
An acute angle is an angle that measures less than 90 degrees. It is smaller than a right angle.
Definition of a triangle
A triangle is a closed figure with three sides and three angles. The sum of all angles in a triangle is always 180 degrees.
Proof
To prove that a triangle must have at least two acute angles, we will consider the possible scenarios for the angles in a triangle.
Scenario 1: All angles are acute
If all three angles of a triangle are acute, then the sum of the angles would be less than 180 degrees. However, the sum of the angles in a triangle is always 180 degrees. Therefore, this scenario is not possible.
Scenario 2: One angle is acute and two angles are obtuse
If one angle of a triangle is acute and the other two angles are obtuse, then the sum of the angles would be greater than 180 degrees. This violates the property that the sum of the angles in a triangle is always 180 degrees. Therefore, this scenario is not possible.
Scenario 3: Two angles are acute and one angle is obtuse
If two angles of a triangle are acute, then the sum of these two angles would be less than 180 degrees. Let's assume the two acute angles are A and B. The sum of these two angles is A + B. Since the sum of all angles in a triangle is 180 degrees, the third angle must be 180 - (A + B). If A + B is less than 180 degrees, then 180 - (A + B) must be greater than 0 degrees. This means that the third angle must also be acute, as it is greater than 0 degrees. Therefore, in this scenario, the triangle has two acute angles.
Conclusion
Based on the analysis of the possible scenarios, we have proved that a triangle must have at least two acute angles.
A triangle is a polygon with three sides and three angles. In order to prove that a triangle must have at least two acute angles, we will use the properties and definitions of triangles.
Definition of an acute angle
An acute angle is an angle that measures less than 90 degrees. It is smaller than a right angle.
Definition of a triangle
A triangle is a closed figure with three sides and three angles. The sum of all angles in a triangle is always 180 degrees.
Proof
To prove that a triangle must have at least two acute angles, we will consider the possible scenarios for the angles in a triangle.
Scenario 1: All angles are acute
If all three angles of a triangle are acute, then the sum of the angles would be less than 180 degrees. However, the sum of the angles in a triangle is always 180 degrees. Therefore, this scenario is not possible.
Scenario 2: One angle is acute and two angles are obtuse
If one angle of a triangle is acute and the other two angles are obtuse, then the sum of the angles would be greater than 180 degrees. This violates the property that the sum of the angles in a triangle is always 180 degrees. Therefore, this scenario is not possible.
Scenario 3: Two angles are acute and one angle is obtuse
If two angles of a triangle are acute, then the sum of these two angles would be less than 180 degrees. Let's assume the two acute angles are A and B. The sum of these two angles is A + B. Since the sum of all angles in a triangle is 180 degrees, the third angle must be 180 - (A + B). If A + B is less than 180 degrees, then 180 - (A + B) must be greater than 0 degrees. This means that the third angle must also be acute, as it is greater than 0 degrees. Therefore, in this scenario, the triangle has two acute angles.
Conclusion
Based on the analysis of the possible scenarios, we have proved that a triangle must have at least two acute angles.
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Prove that a triangle must have atleast two acute angles.? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Prove that a triangle must have atleast two acute angles.? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that a triangle must have atleast two acute angles.?.
Prove that a triangle must have atleast two acute angles.? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Prove that a triangle must have atleast two acute angles.? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that a triangle must have atleast two acute angles.?.
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