Question

ordering angles of a triangle with a variable
triangle abc is shown in the diagram. the lengths of the sides are in terms of the variable n, where ( n geq 4 ). complete the inequality
( m angle ) ( < m angle ) ( < m angle )

Explanation:

Step1: Compare the side lengths

Given \(n\geq4\).
Let's find the values of the side lengths:

• For side \(AC=n + 3\), when \(n = 4\), \(AC=4+3=7\). As \(n\) increases, \(AC\) increases.
• For side \(BC = 2n\), when \(n=4\), \(BC=2\times4 = 8\). As \(n\) increases, \(BC\) increases.
• For side \(AB=3n - 2\), when \(n = 4\), \(AB=3\times4-2=10\). As \(n\) increases, \(AB\) increases.

Now, compare the expressions:
Subtract \(AC\) from \(BC\): \(2n-(n + 3)=n-3\). Since \(n\geq4\), \(2n-(n + 3)=n - 3\geq1>0\), so \(BC>AC\).
Subtract \(BC\) from \(AB\): \((3n-2)-2n=n - 2\). Since \(n\geq4\), \((3n - 2)-2n=n-2\geq2>0\), so \(AB>BC\).

So, \(AC

Step2: Use the angle - side relationship

In a triangle, the larger the side length, the larger the angle opposite it.
The angle opposite \(AC\) is \(\angle B\), the angle opposite \(BC\) is \(\angle A\), and the angle opposite \(AB\) is \(\angle C\).

Since \(AC

Answer:

\(m\angle B