Question

the side lengths of triangle abc are written in terms of the variable p, where p ≥ 3. which is correct regarding the angles of the triangle? m∠a>m∠c>m∠b; m∠b>m∠a>m∠c; m∠c>m∠a>m∠b; m∠c>m∠b>m∠a

Explanation:

Step1: Compare side - lengths

We have side lengths \(AB = 4p - 1\), \(BC=3p\), and \(AC = p + 4\). Since \(p\geq3\), let's find the order of side - lengths.
First, compare \(AB\) and \(BC\):
\(AB - BC=(4p - 1)-3p=p - 1\). Since \(p\geq3\), \(p - 1>0\), so \(AB>BC\).
Next, compare \(BC\) and \(AC\):
\(BC-AC = 3p-(p + 4)=2p-4\). When \(p\geq3\), \(2p - 4=2(p - 2)>0\), so \(BC>AC\).
So the order of side - lengths is \(AB>BC>AC\).

Step2: Use the angle - side relationship in a triangle

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

Answer:

C. \(m\angle C>m\angle A>m\angle B\)