May
29
2 minute read

I worked with my son on this problem last night from AMC8 and I think its really interesting and plays off the previous pentagon problem I had given to the kids in the club.

In the pentagon above $\angle A = 20 ^{\circ}$  The triangle containing A is isosceles. What is $\angle B$ + $\angle D$?

General Steps.

  1. Determine the 2 other angles must be 80$^{\circ}$ since its an isosceles triangle.

  2. Find the supplementary angles on the interior.

  1. Continue tracing the angles for the two triangle B and E since you have one corner already. I assigned x to $\angle B$ and that means its final angle must be 180 - (x + 80) = 100 - x. While I’m at it I added in its supplement on the interior 80 + x.

  1. Repeat the process for $\angle E$ letting its value be y.  Which also derives an angle of 100 - y and a supplement of 80 + y.

  2. Applying the fact that sum of the angle of the interior pentagon are 540 and we now know 4 of the angles, the fifth must be 540 - (100 + 100  + (80 + x) + (80 + y)) =  180 - x - y.  Note: its not a coincidence that the sum of the angle pairs both contain 180 degrees.

  3. Now attack the final triangle spoke anchored by D.  Two of its angles are supplements of ones we’ve already found: 100 - y and x + y.  So $\angle D$   = 180 - ((100 - y) + (x + y)) = 80 - x.

  4. Lo and behold when you add $\angle B$ + $\angle D$ the x cancels out: x + 80 - x = 80.

In fact you can generalize this process to see the relationship between any interior angle of a triangle and 2 exterior spokes of the pentagon.

I think this is one of those interesting inherent relationships and would make a nice progression but I’m not sure if I can cram everything into a worksheet session. To make it work I’d need to:

  • Review basic angle tracing for two intersecting lines and triangles.
  • Review finding the sum of the angles of a polygon at least for a pentagon by breaking it into triangles.
  • Walk through some more abstract angle tracing examples with variables. The symbol manipulation is a little abstract here.
  • Give the problem and tell the kids to start tracing.
  • Be ready to hint about assigning a variable to the two vertices or maybe just supply that from the start.

If I’m not satisfied this will work out I may just go back to my original ideas about doing a intro graph theory problem like the bridges of konigsberg.

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