A triangle having three equal sides is necessary and sufficient for a triangle having three equal angles
Yes
To determine whether a triangle with three equal sides also has three equal angles, we can rely on the concept of congruent triangles. Two triangles are considered congruent when all corresponding sides and angles are equal.
In this case, we can start by assuming that the triangle has three equal sides. This property is known as an equilateral triangle. If we want to prove that it also has three equal angles, we can utilize the following steps:
1. Draw an equilateral triangle and label the three sides as A, B, and C. Since all sides are equal, we can write A = B = C.
2. To calculate the measure of the angles, we can use the fact that the sum of angles in a triangle is always 180 degrees. Let's label the angles as α, β, and γ.
3. Divide the equilateral triangle into three congruent triangles by drawing perpendiculars from the vertices to the opposite sides.
4. Each of these smaller triangles will have a right angle and two acute angles. Since the triangle is equilateral, all the sides of each smaller triangle are equal.
5. We can apply the concept of congruent triangles to the smaller triangles. Due to their congruence, we know that the acute angles in each of the smaller triangles are also equal.
6. Considering the sum of angles in each of the smaller triangles, the right angle is 90 degrees and the sum of the two equal acute angles is 90 degrees. Therefore, each acute angle in the smaller triangles measures 45 degrees.
7. Thus, we can conclude that all three angles in the equilateral triangle are equal and measure 60 degrees.
By following these steps, we have proven that a triangle with three sides of equal length (equilateral triangle) indeed has three equal angles, each measuring 60 degrees. Therefore, having three equal sides is both necessary and sufficient for having three equal angles in a triangle.