Inscribed angle problem [ math/geometry question ]

[adam2016]

Hey guys,

I posted this on a math help sub reddit almost 6 hours ago, and after 6 hours of patiently waiting I think it's safe to say nobody can help me there lol maybe someone on my most visited forum can help out :)

here is the problem ( pictures included )

https://ibb.co/CQYm1Mk

we are told that angle ABC equals 122 degrees, from this we need to find the angle ADC. Obviously ADC will be half the measurement of ABC so right off the bat I know the answer is 61 degrees, but I'm more concerned about how we solve this problem rather than the answer, so I drew a line through the diameter and it looks something like this - https://ibb.co/K07y1dF

as you can see the line doesn't cut the angles in half instead the part closer to A looks much larger than the part closer to C, and it's not just how I have drawn it, the khan academy site draws the line in similar fashion - https://ibb.co/HD1nG7Y

( solving the problem picture ) - https://ibb.co/SQ0c6CL

so since the diameter cuts both angles in two, I named angle ADC Psi and I named angle ABC theta, so as mentioned since the diameter cuts the angles in two I have psi1 and psi2 with theta1 and theta2, by the diagram I drew you can tell that psi1 and theta1 are much larger than their other parts respectively but I decided to pretend they are evenly divided by the diameter. So half of 122 = 61 so theta one = 61. Again as you can see in the picture I algebraically solved for psi1 and I got 1/2 of thera1 which = 30.5.

so by the picture you can see I subbed in 61 for theta1 treating both sides as halves, but in reality theta1 is much larger than theta2, so how would we know what value to sub in as judging by the images one is much larger than the other?

so here is the solution by Khan academy, first divide the angles into twos

angle ABC = angle CBE + angle ABEangle ADC = angle CDE + angle ADE

- https://ibb.co/RTWnGwv

so this makes sense so far.

the pink sides are radii, so the base angles will be the same , angle BCD = angle CDE,

- https://ibb.co/zrZLr92

base angles make sense, but why call it CDE shouldn't it be CDB and not CDE???

the base angles in a triangle add up to 180 so angle CBD + angle BCD + angle CDE equals 180 degrees

- https://ibb.co/Nxh66YV

this makes sense to me as the sum of the interior angles of a triangle will be 180 degrees but again why angle CDE and not CBD??

and the rest which I fail to understand

- https://ibb.co/8ccG6XX

edit* to add some clarity, khan academy says angle BCD + angle CBD + angle CDE = 180??
but how is this true? are angle CDE and angle CDB the same? if so how? also when I sub CDB into the equation instead of CDE I get a completely different answer.

What we do know is that CDE equals BCD, but again how? I thought CDB (not CDE as point E isn't one of the three points that make up the triangle) and BCD would be equal as they are both points in the triangle.

sorry if this is unclear,again the diagrams may explain what I'm trying to express a lot more clearly.


thanks!

[ne555]

> are angle CDE and angle CDB the same? if so how?
yes, same angle
B is inside the segment DE, the ray DB is the same ray as DE

[lastchance]

https://www.mytutor.co.uk/blog/the-circle-theorems-you-need-to-know-and-where-they-come-from/

You need the first theorem.

[adam2016]

thanks guys :) sorry about the late reply, hard drive failed so was offline for a couple days, switched to an SSD ( about time ).