SOLUTION: We are learning similar triangles using AA, SSS and SAS. I am stuck on this problem using similar triangles. Can someone please help? Thanks. If {{{FG/BG = KG/CG}}} ABG=130, a

Question 568916: We are learning similar triangles using AA, SSS and SAS.
I am stuck on this problem using similar triangles. Can someone please help?
Thanks.
If ABG=130, and ADJ=20, find CFD, GKF, GFK, G, and GKJ.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
triangle FGK is similar to triangle BGC by SAS.
FG is proportional to BG in the same ratio that KG is proportional to CG.
angle KGF is congruent to angle BGC because they are the same angle.

since these triangles are similar, their corresponding angles are congruent.
congruent angles have equal measure.

you are given that angle ABG is equal to 130 degrees.
this means that angle GBC is equal to 50 degrees because these angles are supplementary.

angle GFK is the corresponding angle in triangle FGK, so angle GFK is also equal to 50 degrees because the triangles are similar.

since angle GFK is equal to 50 degrees, this means that angle DFC is also equal to 50 degrees because angle GFK and angle DFC are vertical angles and vertical angles are equal.

since angle DFC is equal to 50 degrees and angle FDC is given as being equal to 20 degrees, this means that angle FCD is equal to 110 because the sum of the angles of a triange is equal to 180 degrees.

since angle FCD is equal to 110 degrees, this means that angle GCB is equal to 70 degrees because angle FCD and angle GCB are supplementary.

since angle GCB is equal to 70 degrees, this means that angle GKF is equal to 70 degrees because angle GKF is the corresponding angle in triangle FGK to angle GCB in triangle BGC.

since angle GKF is equal to 70 degrees and angle GFK is equal to 50 degrees, this means that angle KGF is equal to 60 degrees because the sum of the angles of a triangle have to be equal to 180 degrees.

since angle KGF is equal to 60 degrees, this means that angle BGC is also equal to 60 degrees because it's the same angle.

angle GKJ is supplementary to angle GKF which means that angle GKJ is equal to 110 degrees since angle GKJ and angle GKF are supplementary which means that the sum of their angles has to be equal to 180 degrees.

note also that angle GKJ is an exterior angle to triangle GKF which means that it is equal to the sum of the remote interior angles of triangle GKF. these angles are KGF and GFK whose sum is 60 + 50 = 110 degrees.

the following diagram should support the logic just given.
$$$$

in this diagram, triangle FGK was turned around so you can see the corresponding angles and corresponding sides easier.

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