4. Find the range of possible measures of the third side. Use the variable $$ \mathbf{x} $$. $$ 19,13 $$

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To find the range of possible measures for the third side $$ x $$ of a triangle when the other two sides are given as $$ 19 $$ and $$ 13 $$, we 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. Given the sides $$ a = 19 $$ and $$ b = 13 $$, we can set up the following inequalities based on the triangle inequality: 1. $$ a + b > x $$ 2. $$ a + x > b $$ 3. $$ b + x > a $$ Now, let's substitute the values of $$ a $$ and $$ b $$ into these inequalities: 1. $$ 19 + 13 > x $$ 2. $$ 19 + x > 13 $$ 3. $$ 13 + x > 19 $$ Now, we will simplify each inequality: 1. $$ 32 > x $$ or $$ x < 32 $$ 2. $$ x > 13 - 19 $$ or $$ x > -6 $$ (this inequality is always true since $$ x $$ must be positive) 3. $$ x > 19 - 13 $$ or $$ x > 6 $$ Now, combining the valid inequalities, we have: - From the first inequality: $$ x < 32 $$ - From the third inequality: $$ x > 6 $$ Thus, the range of possible measures for the third side $$ x $$ is: $$ 6 < x < 32 $$ This means that the third side must be greater than $$ 6 $$ and less than $$ 32 $$. The third side $$ x $$ must be greater than 6 and less than 32.

4. Find the range of possible measures of the third side. Use the variable \( \mathbf{x} \). \[ 19,13 \]

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