Let's analyze the question step-by-step, focusing on the geometrical properties of the triangles:
1. Understanding the sides and angles:
- We have three triangles formed by points [tex]\(U\)[/tex], [tex]\(V\)[/tex], [tex]\(S\)[/tex], and [tex]\(R\)[/tex] along a common vertex [tex]\(T\)[/tex].
- The sides opposite angles [tex]\(UTV\)[/tex], [tex]\(UTS\)[/tex], and [tex]\(STR\)[/tex] are [tex]\(UV\)[/tex], [tex]\(US\)[/tex], and [tex]\(SR\)[/tex] respectively.
2. Given relationship between sides:
- The side lengths are such that [tex]\( UV > US > SR \)[/tex].
- Knowing this, we can infer the relationship between the angles of the triangles formed.
3. Relationship between side lengths and angles in a triangle:
- In any triangle, the larger the side, the larger the angle opposite to that side.
- This is called the Triangle Inequality Theorem.
Given the triangles and their side lengths:
- [tex]\( UV \)[/tex] is the longest side, hence [tex]\( \angle UTV \)[/tex] is the smallest angle.
- [tex]\( US \)[/tex] is shorter than [tex]\( UV \)[/tex] but longer than [tex]\( SR \)[/tex], hence [tex]\( \angle UTS \)[/tex] is larger than [tex]\( \angle UTV \)[/tex] but smaller than [tex]\( \angle STR \)[/tex].
- [tex]\( SR \)[/tex] is the shortest side, hence [tex]\( \angle STR \)[/tex] is the largest angle.
Therefore, the true statement about the angles is:
- [tex]\( m \angle UTV < m \angle UTS < m \angle STR \)[/tex].
So, the correct statement is:
[tex]\[ \boxed{m \angle UTV < m \angle UTS < m \angle STR} \][/tex]
