<p>In triangle ABC above, if a>b>c, which of the following must be true?</p>
<p>I. 60<a<180
II. 45<b<90
III. 0<c<60</p>
<p>A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III</p>
<p>The figure is unnecessary; it's just a triangle with angles a, b, and c. I can imagine why C is correct, but how can I approach it algebraically (using inequalities, probably?) ?</p>
<p>I’m not sure Algebra is the hammer you want for that nail. Logic seems more appropriate.</p>
<p>I. a is always going to be less than 180, because the sum of the three angles is 180. As the largest of three angles, a also has to be greater than 60 (180/3)</p>
<p>III. c is always going to be greater than zero, because the figure is a triangle and not a line. As the smallest of three angles, c also has to be less than 60 (180/3)</p>
<p>II. is the only option that requires significant thought. b does have to be less than 90, because otherwise a+b > 180, and you’re out of degrees before you even get to c. But b does not have to be greater than 45, because b = 44, c = 43, a = 93 is a counterexample.</p>
<p>So, say I have a<b<c<d. And a+b+c+d = 200.</p>
<p>d > 50 (200/4), and a < 50, but we can’t say anything about b and c?</p>
<p>If a,b,c,d are nonnegative, you can say that b < 200/3 (see why?). By a similar argument, c < 100. But that’s about all you can say about b and c.</p>