## A lovely angle puzzle

I’ve written before about the app “Brilliant“, which is well worth getting, and I also follow their Facebook page which provides me with a regular stream questions. Occasionally I have to think about how to tackle them, and they’re excellent. More often, a question comes up that I look at and think would be awesome to use in a lesson.

Earlier this week this question popped up:

What a lovely question that combines algebra and angle reasoning! I can’t wait to teach this next time, and I am planning on using this as a starter with my y11 class after the break.

The initial question looks simple, it appears you sum the angles and set it equal to 360 degrees, this is what I expect my class to do. If you do this you get:

*7x + 2y + 6z – 20 = 360*

*7x + 2y + 6z = 380 (1)*

I anticipate some will try to give up at this point, but hopefully the resilience I’ve been trying to build will kick in and they’ll see they need more equations. If any need a hint I will tell them to consider vertically opposite angles. They should then get:

*2x – 20 = 2y + 2z (2)*

*And*

*3x = 2x + 4z (3)*

I’m hoping they will now see that 3 equations and 3 unknowns is enough to solve. There are obviously a number of ways to go from here. I would rearrange equation 3 to get:

*x = 4z (4)*

Subbing into 2 we get:

*8z – 20 = 2y + 2z*

*6z = 2y + 20 (5)*

Subbing into 1

*28z + 2y + 6z = 380*

*34z = 380 – 2y (6)*

Add equation (5) to (6)

*40z = 400*

*z = 10 (7)*

Then equation 4 gives:

*x = 40*

And equation 2 gives:

*60 = 2y + 20*

*40 = 2y*

*y = 20.*

From here you can find the solution x + y + z = 40 + 20 + 10 = 70.

A lovely puzzle that combines a few areas and needs some resilience and perseverance to complete. I enjoyed working through it and I’m looking forward to testing it out on some students.

*Cross-posted to Betterqs here.*

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### Comments welcome......

A nice puzzle. Surely you can solve it more simply using the 3x and 2x-20 and angles on a straight line, to give 5x-20=180 so X=40 then use the 2x-20 and 2x+4z add up to 180 to solve for z, (80-20+80+4z=180, so z=10) and then the remaining 2y+2z and 2x+4z to solve for y (2y+20+80+40=180, so y=20). So no need for simultaneous equations at all?

I’ve missed out some working out for the sake of the typing, but would expect it in class.

Aye, my brains default seems to always be to overcomplicate matters. The liner pair straight lines work out a much more concise method. A good discussion point in a lesson if people look at different ways.

Not seen “Brilliant” before, thanks for calling it to my attention. Nice problem, I’ll try it on my Y11 but suspect they will need help remembering what an angle is!

Haha, do let me know how they get on!