Acute Angles in a Triangle

Date: 12/02/98 at 19:46:36
From: Kailey van der Spank
Subject: Math - Grade 4 Triangles

My question is: What is the greatest number of angles smaller than a 
right angle a triangle can have? I said 2 and got the answer wrong. 
Help.

Date: 12/03/98 at 13:05:41
From: Doctor Peterson
Subject: Re: Math - Grade 4 Triangles

Hi, Kailey. I think you may have just read the question backward, 
because your answer would be right if the question were a little 
different.

If one angle of a triangle is obtuse (bigger than a right angle), then 
the others both have to be acute (less than a right angle). You can 
either see that by just drawing an obtuse angle and seeing what happens 
if you make a second angle obtuse, or by knowing that the sum of the 
angles is always 180 degrees. This means that you have to have at least 
two acute angles.

    \                                /
     \                              /
      \                            /
       \                          /
        \                        /
         \                      /
          \                    /
           \                  /
            \                /
             +--------------+

But the question is not the least number of acute angles, but the 
greatest. If you draw a simple, ordinary triangle, it is likely to have 
three acute angles. For instance, an equilateral triangle will work:

                    +
                   / \
                  /   \
                 /     \
                /       \
               /         \
              /           \
             /             \
            +---------------+

So a triangle can have either 2 or 3 acute angles, and the maximum is 
3, not 2.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/