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 \(a=p + 4\), \(b = 3p\), \(c=4p-1\) with \(p\geq3\).
First, find the difference between \(b\) and \(a\): \(b - a=3p-(p + 4)=2p-4\). Since \(p\geq3\), \(2p-4=2(p - 2)\geq2(3 - 2)=2>0\), so \(b>a\).
Next, find the difference between \(c\) and \(b\): \(c - b=(4p-1)-3p=p - 1\). Since \(p\geq3\), \(p - 1\geq2>0\), so \(c>b\).
So the order of side - lengths is \(c>b>a\).

Step2: Use the angle - side relationship in a triangle

In a triangle, the larger the side - length, the larger the angle opposite it.
Since \(c\) is opposite \(\angle B\), \(b\) is opposite \(\angle A\), and \(a\) is opposite \(\angle C\), we have \(m\angle C>m\angle A>m\angle B\).

Answer:

\(m\angle C>m\angle A>m\angle B\) (the option that reads \(m\angle C>m\angle A>m\angle B\))