Given triangle ABC with an angle of 20 degrees at A bisected by AF and an angle of 30 degrees at B cut by a segment BE such that angle ABE is 10 degrees and angle CBE is 20 degrees. What is the measure of angle DCE?

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Discussion

Hints: Warning, these may not be helpful!!! GSP files are provided.

Hint 1

Hint 2Now can we show that the added lines to this figure embed the original problem?

Hint 3Since we know that AD = DB, take D as the center of a circle and extend lines from the triangle to explore sizes of subtended angles:

It is very tempting to see the measure of Arc IJ to be 20 degrees and hence the measure of arc IL to be 120 degees. That would mean that angle CDE is 120 degrees and therefore in triangle CDE we would know two of the angles, 120 degrees and 30 degrees, so the other angle, Angle DCE, is 30 degrees. HOWEVER, can we prove arc IJ has a measure of 20 degrees?