Challenging problem corrrection no sine and no cosine law

[davedave]

sorry about the mistake in my last post. I miswrote the bottom vertex of the equilateral triangle.

Let me re-state the problem correctly

This is the 3rd and final question I post from the book, The Unsolvable and the Solvable.

It is NOT a homework question. This is something for everyone to try out for interest.

Consider an isoceles triangle ABC and an equilateral triangle BCF which share the side BC as shown below. Please ignore the dotted lines.

......A


......D
.........E

....B_________________C



......F

D is a point on side AB and E is a point on side AC.

angle DAE=20 degrees
angle DEA=20 degrees
angle EDC=10 degrees
angle ECD=10 degrees
angle DBC=80 degrees
angle DCB=70 degrees
angle BDC=30 degrees

WITHOUT using the sine and cosine law, determine angle EFC.

 

[Russell Berty]

Label the following unknown angles:

a = BDF b = FDC c = DEF d = FEC e = BFD f = DFE g = EFC

Then, write 7 different equations involving them.

For example, a + e + 140 = 180.

Once you have 7 equations with 7 unknowns, it can be solved. Though, it is messy. If you know linear algebra, I would use matrices to solve the system.