Questions: In Δ STU, TU=17, US=12, and ST=15. Which statement about the angles of Δ STU must be true?

Transcript text: In $\Delta \mathrm{STU}, \mathrm{TU}=17, \mathrm{US}=12$, and $\mathrm{ST}=15$. Which statement about the angles of $\Delta \mathrm{STU}$ must be true?

Solution

Solution Steps

To determine the relationship between the angles of triangle $\Delta \mathrm{STU}$, we can use the fact that in any triangle, the side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest. Given the side lengths $\mathrm{TU}=17$, $\mathrm{US}=12$, and $\mathrm{ST}=15$, we can compare these lengths to determine the order of the angles.

Step 1: Identify the Side Lengths

In triangle $\Delta \mathrm{STU}$, the side lengths are given as follows:

$\mathrm{TU} = 17$
$\mathrm{US} = 12$
$\mathrm{ST} = 15$

Step 2: Determine the Order of the Angles

To find the relationship between the angles, we compare the lengths of the sides. The side opposite the smallest angle is the shortest side, and the side opposite the largest angle is the longest side.

From the side lengths:

The shortest side is $\mathrm{US} = 12$ (opposite $\angle T$)
The middle side is $\mathrm{ST} = 15$ (opposite $\angle U$)
The longest side is $\mathrm{TU} = 17$ (opposite $\angle S$)

Step 3: Establish the Angle Inequalities

Based on the side lengths, we can establish the following inequalities for the angles: \[ \mathrm{m} \angle T < \mathrm{m} \angle U < \mathrm{m} \angle S \]