In a triangle, the lengths of the sides are indeed related to the measures of the angles. The Triangle Inequality Theorem helps to understand this relation. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Additionally, there is a relationship between the lengths of the sides of a triangle and the measures of its angles: the longest side of a triangle is always opposite the largest angle, and conversely, the smallest angle is always opposite the shortest side. This is because the larger the angle, the longer the side across from it needs to be to close the triangle.
Based on this information, let's analyze the provided statements to determine which is true:
A. The longest side length is opposite the largest angle measure.
This statement is true. With triangles, the largest angle is directly opposite the longest side due to the principles mentioned above.
B. The shortest side length is opposite the largest angle measure.
This statement is false. The shortest side cannot be opposite the largest angle since the longest side is opposite the largest angle, as previously explained.
C. The shortest side length is opposite the second largest angle measure.
This statement is generally false because the shortest side is opposite the smallest angle, not necessarily the second largest.
D. The side lengths and angle measures of a triangle are not related.
This statement is false. The side lengths and angle measures of a triangle are related through both the Triangle Inequality Theorem and the principle relating angles to their opposite sides.
Therefore, the correct statement about the sides of a triangle is:
The longest side length is opposite the largest angle measure.
