if a triangle has three equal sides, what is true about a triangle? A. all three angles must be equal. B. all three angles must be unequal. C. one angle must be a right angle. D. two angles must be 45-degree angles.
The correct answer and explanation is:
The correct answer is A. all three angles must be equal.
When a triangle has three equal sides, it is called an equilateral triangle. By definition, all sides in an equilateral triangle have the same length. This equality in side lengths leads to a specific property in the angles of the triangle.
In any triangle, the angles opposite to equal sides are also equal. Since all three sides are equal in an equilateral triangle, all three angles must be equal as well. The sum of the interior angles in any triangle is always 180 degrees. Therefore, when the three angles are equal, each angle must measure exactly 60 degrees. This means that an equilateral triangle is also equiangular, having three angles of the same measure.
This property differentiates the equilateral triangle from other types of triangles such as isosceles and scalene triangles. An isosceles triangle has at least two sides equal and the two opposite angles to those sides are equal, but the third angle is generally different. A scalene triangle has all sides of different lengths and consequently all angles are different.
Options B, C, and D are incorrect because they do not describe the properties of a triangle with three equal sides. B states that all angles must be unequal, which contradicts the equality of sides and angles in an equilateral triangle. C says one angle must be a right angle, which is only true for right triangles, not equilateral ones. D mentions two angles must be 45 degrees, which would make the triangle a right isosceles triangle, not an equilateral one.
In summary, having three equal sides in a triangle guarantees that all three angles are also equal, each measuring 60 degrees, making the triangle equilateral and equiangular.