To apply the Triangle Inequality Theorem, we need to consider the relationships between the lengths of the sides of the triangle. The theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
In this case, let's denote the lengths of the two known sides as \(a = 6\) and \(b = 2\), and let the length of the third side be \(c\). Now we will apply the triangle inequality to find the possible values for \(c\).
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From \(a + b > c\): \[ 6 + 2 > c \implies 8 > c \implies c < 8 \]
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From \(a + c > b\): \[ 6 + c > 2 \implies c > 2 - 6 \implies c > -4 \] Since \(c\) must be a positive whole number, this condition doesn't restrict \(c\) further.
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From \(b + c > a\): \[ 2 + c > 6 \implies c > 6 - 2 \implies c > 4 \]
Now we can combine the results from the inequalities:
- From the first inequality, we have \(c < 8\).
- From the second inequality (which is always true for positive \(c\)), we don't need to consider it further.
- From the third inequality, we have \(c > 4\).
Combining these results, we find: \[ 4 < c < 8 \]
Since \(c\) must be a whole number, the possible values of \(c\) are \(5, 6, 7\).
Listing these in ascending order, the measure of the third side could be 5, 6, or 7.
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