Here is a suggested method of teaching this, which stresses the idea of starting with the concrete and encouraging self-discovery.

 

We have a right triangle, one angle 45°. What is the other angle? (It must be 45° also.) What type of triangle is it (isosceles)? Given a side a, what is the hypotenuse (√2a)?

 

Given a right triangle with a = ½ h. What is the other side? There are two angles: 30° and 60°. Which is the 30°?

 

We now have two special triangles: 45°, and a 30°-60° triangle. Can you think of any others? One student said 60°, 70°, and 80°. Wrong, as angles must add to 180°. Also one angle must be 90°.

 

Now that we have discovered these special triangles, we can use this knowledge in helping us solve problems.

 

Another example: The 3,4,5 triangle is special, as the three sides are integers. The 6,8,10 also has three sides which are integers. What can you say about these two triangles (are similar)? Okay, we have one special triangle, along with all similar triangles. The 5,12,13 is also special. Knowing this may make it easier to solve problems.

 

A student asked how does one calculate tan(40°)? One way is to write it as tan(45°-5°). We know the value of tan(45°). We then use approximate methods which the class did not learn yet.