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Can an acute-angled triangled be a. and ...

Can an acute-angled triangled be a. and equilateral, b. an isosceles, and c. a scalene triangle ?

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To determine whether an acute-angled triangle can be equilateral, isosceles, or scalene, let's analyze each type of triangle step by step. ### Step 1: Understanding Acute-Angled Triangle An acute-angled triangle is defined as a triangle where all three angles are less than 90 degrees. ### Step 2: Equilateral Triangle - **Definition**: An equilateral triangle has all three sides of equal length and all three angles equal. - **Angles**: Since the sum of angles in any triangle is 180 degrees, if all angles are equal in an equilateral triangle, each angle must be: \[ \text{Angle} = \frac{180}{3} = 60 \text{ degrees} \] - **Conclusion**: Since 60 degrees is less than 90 degrees, an equilateral triangle is indeed an acute-angled triangle. ### Step 3: Isosceles Triangle - **Definition**: An isosceles triangle has at least two sides of equal length, which means at least two angles are equal. - **Assumption**: Let’s assume the two equal angles are \( A \) and \( B \). - **Angle Sum Property**: According to the angle sum property: \[ A + B + C = 180 \text{ degrees} \] Since \( A = B \), we can write: \[ 2A + C = 180 \text{ degrees} \] - **Condition for Acute Angles**: If \( A < 90 \) degrees, then: \[ 2A < 180 \text{ degrees} \] This implies: \[ C = 180 - 2A \] Since \( A \) is less than 90 degrees, \( C \) will also be less than 90 degrees for many values of \( A \) (for example, if \( A = 60 \) degrees, then \( C = 60 \) degrees). - **Conclusion**: Therefore, an isosceles triangle can also be an acute-angled triangle. ### Step 4: Scalene Triangle - **Definition**: A scalene triangle has all sides of different lengths, which means all angles are different. - **Angle Sum Property**: Again, using the angle sum property: \[ A + B + C = 180 \text{ degrees} \] - **Example**: Let’s take an example where \( A = 40 \) degrees, \( B = 60 \) degrees, and \( C = 80 \) degrees. - All angles are less than 90 degrees. - **Conclusion**: Hence, a scalene triangle can also be an acute-angled triangle. ### Final Conclusion An acute-angled triangle can indeed be: - a) An equilateral triangle - b) An isosceles triangle - c) A scalene triangle
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