oliviarubinstein0
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A farmer is building three triangular pens such that [tex]\(\overline{VT}\)[/tex], [tex]\(\overline{UT}\)[/tex], [tex]\(\overline{ST}\)[/tex], and [tex]\(\overline{RT}\)[/tex] are line segments. If [tex]\(UV \ \textgreater \ US \ \textgreater \ SR\)[/tex], which is a true statement?

A. [tex]\(m \angle UTV = m \angle UTS = m \angle STR\)[/tex]
B. [tex]\(m \angle UTV \ \textless \ m \angle UTS \ \textless \ m \angle STR\)[/tex]
C. [tex]\(m \angle UTS \ \textgreater \ m \angle UTV \ \textgreater \ m \angle STR\)[/tex]
D. [tex]\(m \angle UTV \ \textgreater \ m \angle UTS \ \textgreater \ m \angle STR\)[/tex]

Sagot :

GinnyAnswer
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]
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