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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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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