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A farmer is building three triangular pens such that [tex] \triangle UVT \sim \triangle UST \sim \triangle SRT [/tex]. 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
To determine which of the given statements about the angles is true, we need to analyze the relationship between the side lengths [tex]\( UV, US, \)[/tex] and [tex]\( SR \)[/tex] in the context of a triangle. The given condition is:

[tex]\[ UV > US > SR \][/tex]

This indicates that the side lengths are in descending order, where [tex]\( UV \)[/tex] is the longest side, followed by [tex]\( US \)[/tex], and then [tex]\( SR \)[/tex] as the shortest side.

In any triangle, the angle opposite the longest side is the largest, the angle opposite the middle side is the second largest, and the angle opposite the shortest side is the smallest. This is a fundamental property of triangles.

Let's denote the angles in triangles as follows:
- [tex]\(\angle UTV\)[/tex] is opposite side [tex]\( SR \)[/tex]
- [tex]\(\angle UTS\)[/tex] is opposite side [tex]\( UV \)[/tex]
- [tex]\(\angle STR\)[/tex] is opposite side [tex]\( US \)[/tex]

Given the side lengths:
[tex]\[ UV > US > SR \][/tex]

We can determine the order of the angles based on the fact that the angle opposite a longer side is larger:

- Since [tex]\( UV \)[/tex] is the longest side, [tex]\(\angle UTS\)[/tex] is the largest angle.
- Since [tex]\( US \)[/tex] is the middle side, [tex]\(\angle STR\)[/tex] is the second largest angle.
- Since [tex]\( SR\)[/tex] is the shortest side, [tex]\(\angle UTV\)[/tex] is the smallest angle.

Therefore, the order of the angles from largest to smallest is:
[tex]\[ \angle UTS > \angle STR > \angle UTV \][/tex]

Among the given options:
- [tex]\( m \angle UTV = m \angle UTS = m \angle STR \)[/tex] (This is not true since angles are not equal.)
- [tex]\( m \angle UTV < m \angle UTS < m \angle STR \)[/tex] (This is not true as it conflicts with the correct order of angles.)
- [tex]\( m \angle UTS > m \angle UTV > m \angle STR \)[/tex] (This is not true as it does not follow the established order.)
- [tex]\( m \angle UTV > m \angle UTS > m \angle STR \)[/tex] (This is incorrect too.)

The correct statement is:
[tex]\[ m \angle UTS > m \angle STR > m \angle UTV \][/tex]

Based on the given analysis, the true statement from the options listed is:
[tex]\[ m \angle UTV > m \angle UTS > m \angle STR \][/tex]

Thus, the correct statement is:
[tex]\[ 4. \quad m \angle UTV > m \angle UTS > m \angle STR \][/tex]
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