Question

list the angle measures of $\triangle uvw$ in order from smallest to largest. assume that $z$ is a positive number.

$m\angle\square < m\angle\square < m\angle\square$

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the length of a side, the larger the measure of the opposite angle. So we first need to compare the lengths of the sides: \(44z\), \(64z\), and \(65z\). Since \(z>0\), we can compare the coefficients directly. We have \(44 < 64 < 65\), so \(44z<64z < 65z\).

Step2: Identify the opposite angles

• The side of length \(44z\) is opposite \(\angle U\).
• The side of length \(64z\) is opposite \(\angle V\).
• The side of length \(65z\) is opposite \(\angle W\).

Step3: Determine the order of angles

Since the side opposite \(\angle U\) (\(44z\)) is the shortest, \(\angle U\) is the smallest angle. The side opposite \(\angle V\) (\(64z\)) is longer than \(44z\) but shorter than \(65z\), so \(\angle V\) is larger than \(\angle U\) but smaller than \(\angle W\). The side opposite \(\angle W\) (\(65z\)) is the longest, so \(\angle W\) is the largest angle.

Answer:

\(m\angle U