Determining triangle type from three integer inputs
You could encapsulate the conditions in a PotentialTriangle class. Note that the fields are of type long to avoid overflow when calculating whether triangle inequality is violated.

public class PotentialTriangle {
    private final long a, b, c;

    public PotentialTriangle(int sideA, int sideB, int sideC) {
        a = sideA;
        b = sideB;
        c = sideC;
    }

    public boolean isAnySideTooShort() {
        return a &lt;= 0 || b &lt;= 0 || c &lt;= 0;
    }

    public boolean violatesTriangleInequality() {
        return a &gt; b + c || b &gt; a + c || c &gt; a + b;
    }

    public boolean areSidesEqual() {
        return a == b &amp;&amp; b == c;
    }

    public boolean areAtLeastTwoSidesEqual() {
        return a == b || b == c || c == a;
    }
}


This would vastly simplify your enum selection code, which I would put directly into the enum:

public enum TriangleType {
    ISOSCELES,
    EQUILATERAL,
    SCALENE;

    public static TriangleType ofPotentialTriangle(PotentialTriangle triangle) {
        throwIf(triangle.isAnySideTooShort(), 
                "Length of sides cannot be equal to or less than zero");
        throwIf(triangle.violatesTriangleInequality(), 
                "Sum of any two sides must be larger than the remaining side");

        if (triangle.areSidesEqual()) {
            return EQUILATERAL;
        }
        if (triangle.areAtLeastTwoSidesEqual()) {
            return ISOSCELES;
        }
        return SCALENE;
    }

    private static void throwIf(boolean condition, String message) {
        if (condition) {
            throw new IllegalArgumentException(message);
        }
    }
}