The problem gives an isosceles triangle ABC with base angles of measure 80 and vertex angle of 20. A ray from the base angle C divides the angle into a 60 degree angle and a 20 degree angle. It intersects AB at L. A ray from the base angle at B divides the angle into a 30 degree angle and a 50 degree angle. It intersects AC at K. Draw KL. What is the measure of angle KLC? (Another version of the problem asks for the measure of angle KLB)

Construct the perpendicular to AC from L. Let F be the intersection point on AC. Contruct the angle bisector of angle KCB and let the bisector intersect FL and J, AB at M, and KB at N. Angle NCB = 40 degrees and angle NCL = 20 degrees. Draw KM and KJ.

Now, we have nearly enough relationships to finish the problem.

Triangle ALC is isoscles because the base angles are of the same measure.

Angle CKB = 50 degrees by the triangle angle sum theorem.

Triangle KCB is isosceles because the base angle are the same measure.

CJ is the perpendicular bisector of KB because it is the altitude on an isosceles triangle.>

Triangle KNM is congruent to triangle BNM by SAS.

Fill in some angle measures we know:

Angle MKN and angle MBN each measure 30 degees; Angle NMB and angle NMK have the same measure from congruent triangles.

Angle NMB and angle NMK each measure 60 degrees from triangle angle sum theorem.

Angle LMK has measure of 60 degrees from straight angle along AB.

Angle LMJ has measure of 60 degrees from vertical angles.

FL is a perpendicular bisector of AC because

J is on FL so it is equidistant from A and C. Therefore triangle AJC is isosceles. Hence, LAJ has measure of 20 degrees, same as the measure of angle LCM.

AB is the angle bisector of angle CAJ.

Triangle AJM is congruent to triangle AKM by ASA.

Therefore AKMJ is a kite.

KJ and AM are the diagonals of the kite. Therefore AM is perpendicular to KJ.

Triangles KMR and JMR are each 30-60-90 triangles with common side MR. Therefore they are congruent.

Angle ALF has measure of 70 degrees by triangle angle sum theorem. Therefore angle RLJ has measure of 70 degrees as a vertical angle. Angle LJR has measure of 20 degrees by triangle angle sum theorem.

KR = RJ from congruent triangles KRM and RJM.

Triangle LRJ is conruent to triangle LRK by SAS and hence angle LKR has measure of 20 degrees.

By the triangle angle sum theorem in triangle CKL, angle CKL has a measure of 30 degrees.