Question

angle inequalities
the side lengths of triangle abc are written in terms of the variable p,
where ( p geq 3 )
which is correct regarding the angles of the triangle?
( m angle c > m angle a > m angle b )
( m angle c > m angle b > m angle a )
( m angle b > m angle a > m angle c )
( m angle a > m angle c > m angle b )

Explanation:

Step1: Compare side lengths

Since \(p\geq3\),
For \(p + 4\) and \(3p\):
\(3p-(p + 4)=2p-4\geq2\times3 - 4=2>0\), so \(3p>p + 4\).
For \(4p-1\) and \(3p\):
\(4p-1-3p=p - 1\geq3-1 = 2>0\), so \(4p-1>3p\).
So the side - length order is \(AB=4p - 1>BC = 3p>AC=p + 4\).

Step2: Use the angle - side relationship

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

Answer:

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