Find the range of possible measures of the third side. \[ 13,21 \]

To find the range of possible measures for the third side of a triangle when given two sides of lengths 13 and 21, you can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. 1. The first inequality is: \( 13 + 21 > x \) which simplifies to \( x < 34 \). 2. The second inequality is: \( 13 + x > 21 \) which simplifies to \( x > 8 \). 3. The third inequality is: \( 21 + x > 13 \) which simplifies to \( x > -8 \) (this does not provide any new information since we already have \( x > 8 \)). Combining these inequalities, we find that the range of the possible measures for the third side \( x \) is: \[ 8 < x < 34 \] Therefore, the third side must be greater than 8 and less than 34.