Find the range of possible measures of che third side. Use the variable \( x \). \[ 15,24 \]

To find the range of possible measures for the third side \( x \) of a triangle given the other two sides, you can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For your sides of 15 and 24, we set up the inequalities: 1. \( 15 + 24 > x \) which simplifies to \( x < 39 \). 2. \( 15 + x > 24 \) which simplifies to \( x > 9 \). 3. \( 24 + x > 15 \) which simplifies to \( x > -9 \) (this is always true for positive lengths). Thus, the range for \( x \) is \( 9 < x < 39 \). In practice, understanding this range can help you ensure that any potential triangle you construct adheres to the fundamental properties of triangles, avoiding the dreaded degenerate triangle where the three points are collinear. So next time you're dealing with triangle sides, keep this nifty rule in your back pocket! If you want to dive deeper into the world of triangles, consider exploring types like isosceles or scalene, or even delve into the Pythagorean theorem for right triangles. There’s a whole universe of geometry waiting with facts, riddles, and even famous results like the Law of Sines and Law of Cosines!