Question

list the side lengths of $\triangle stu$ in order from shortest to longest, given that $m\angle s = 5p - 63^\circ$, $m\angle u = 5p - 46^\circ$, and $m\angle t = 2p + 1^\circ$.\
$\square < \square < \square$\
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Explanation:

Step1: Sum angles to 180°

The sum of angles in a triangle is $180^\circ$, so:
$$(5p - 63) + (5p - 46) + (2p + 1) = 180$$

Step2: Simplify and solve for $p$

Combine like terms and isolate $p$:
$$12p - 108 = 180$$
$$12p = 288$$
$$p = 24$$

Step3: Calculate each angle

Substitute $p=24$ into each angle expression:

• $m\angle S = 5(24) - 63 = 120 - 63 = 57^\circ$
• $m\angle U = 5(24) - 46 = 120 - 46 = 74^\circ$
• $m\angle T = 2(24) + 1 = 48 + 1 = 49^\circ$

Step4: Match sides to opposite angles

In a triangle, the shortest side is opposite the smallest angle, and the longest side is opposite the largest angle:

• Smallest angle $\angle T$ → opposite side $SU$
• Middle angle $\angle S$ → opposite side $TU$
• Largest angle $\angle U$ → opposite side $ST$

Answer:

$SU < TU < ST$