Question

@watashiwa
Please look at this solution for your "Find the shortest side the figure problem.

Answer 1

@ Watashiwa

In a triangle:
The shortest side is always opposite the smallest interior angle
The longest side is always opposite the largest interior angle

Note that the theorem says begins "In a Triangle" so work in one triangle at the time.
In the diagram shown in your problem, there are two triangles.

Look at only Triangle BCD. In that triangle, look for the smallest angle which is 55. The side opposite that angle of 55 is side BC with measure BC.

Then, look for the next largest angle in Triangle BCD. That is the 60 angle. The side length opposite it is BD. It will be larger than BC.

The largest angle of Triangle BCD is 65. Opposite it is the side with length CD.

For triangle BCD, the side lengths are related in this way:

BC < BD < CD.

Forget about that for now and look at triangle ABD. Find the smallest angle (50) and look for the side opposite it (BD). Continue as in the previous triangle. You will see that
BD < AB < AD

Now, look at the two inequalities. Look for a common length. BD is in both.

BC < BD < CD and BD < AB < AD.

BC is the smallest side in triangle BCD and BC is also smaller than the smallest side of triangle ABD.

That makes BC the smallest length in the diagram.

Let me know if there are questions. I don't want you to fail Geometry class and will help you as will others here. All you have to do is to ask.

Answer 2

Thank you so much for your in depth explanation! I'm kind of late to seeing this, but it makes way more sense down than just writing down BC on my worksheet! Thank you :) !

Answer 3

Glad to help.