\( 10 n: 25 \) \( m \angle W X Y: \) 66 label optional \( 14 \mathrm{~m} \angle X Y \mathrm{~F}: \) 24 10 Based on the measurements provided, which of the following lista the side measures from smallest to largest? 10) \( \overline{\mathrm{FW}}, \overline{W X}, \overline{X Y} \) 10 \( \overline{W X}, \overline{Y W}, \overline{X Y} \) (0) \( \overline{W X}, \overline{X Y}, \overline{Y W} \) 10 \( \overline{X Y}, \overline{W X}, \overline{Y W} \)

To determine the order of side measures from smallest to largest, we first need to interpret the values and proportions given. With the ratio \(10 n: 25\) and the angle measures \(m \angle W X Y\) at \(66^\circ\) along with \(14 m \angle XYF\) at \(24^\circ\), we can analyze how these relate to triangle properties and the sides opposite these angles. Now, if we consider the angles provided, the side lengths opposite smaller angles in a triangle are shorter compared to those opposite larger angles. That means \( \overline{XY} \) is opposite the smallest angle, and \( \overline{FW} \) is opposing the largest, which helps us figure out our order. Thus, if we look at the angle measures carefully, it’s clear the order of side lengths from smallest to largest is indeed \( \overline{X Y}, \overline{W X}, \overline{Y W} \). Triangles are fascinating shapes that date back to ancient civilizations. The Greeks, particularly through the work of Euclid around 300 BC, laid the foundations for geometric principles involving angles, sides, and ratios, leading to the development of laws like the Law of Sines and the Law of Cosines. These helped mathematicians and architects create structure and art throughout history, even influencing modern engineering and design! If you're looking to dive deeper into understanding how angles and sides correlate in triangles, “Geometry: Euclid and Beyond” by Robin Hartshorne provides an insightful excursion into classical geometry. It’s not just about the rules; it's about the beauty and logic that underpin the very fabric of our spatial world!