# CAS-ing a triangle

Check out this fun little problem from @daveinstpaul.

I’m sure there is a much more elegant solution, but given my technology interests, I thought this would be a cool way to incorporate CAS.  Based on Dave’s restrictions, the following angle measures apply.

There are many ways to write equations from this setup, several of which are identical forms of the same information.  One way to keep from writing dependent equations is to use combinations.  Using only the smallest triangles, you get

2A+Z=180
2X+Y=180
2W+V=180

From the top two triangles, one relationship is  A+Y=180. Because the large triangle is isosceles, the bottom two triangles give X+V=W.  There’s no convenient way to combine the top and bottom triangle.

Combining all three triangles, one relationship is X+W+Z=180.

That’s a system of 6 equations in 6 variables which Wolfram Alpha solves to give A=45 degrees.

I’d love to see a non-algebraic approach.

### 2 responses to “CAS-ing a triangle”

1. Doug Kuhlmann

A bit simpler algebraic approach. A=V since both are vertex angles of isosceles triangles with same base angles. Also, A=X+X since A is an external angle to triangle EBD. W=A+X=3A/2 Finally 180=A+2W=4A. Hence, A=45

2. chrisharrow

While elegance and brevity will always have high value in mathematics, one of the things I most appreciate about student access to CAS is the technology’s ability to support correct student reasoning. The key for students is learning how to ask good questions. What is not necessary is asking tight, efficient questions on the front end. I think many students who struggle with math worry why their thoughts are never as direct or efficient as those their teachers or their textbooks produce. For those who know how to ask a math question algebraically, a CAS can help.
For what it’s worth, your equations are much better than those I pitched. Thanks for your contributions.