April 10, 2016Michael SunAngle ChasingAngle chasing is an extremely important skill. Here is a warm-up:TrianglesABCandADCare isosceles withAB=BCandAD=DC. PointDis inside4ABC,6ABC= 40◦, and6ADC= 140◦. What is the degree measure of6BAD?Source: AMC 12, 2007Solution:ACBD6ABC= 40 implies that6BAC=6BCA= 70.6ADC= 140 implies that6DAC=6DCA= 20.So6BAD= 70-20 =50.Hopefully these types of problems are pretty natural. In general, isosceles and equilateraltriangles are great for angle chasing. Next, you should be familiar with the following fromgeometry:OBACProve that6ABC=126AOC.Proof:As hinted above, isosceles triangles are great for angle chasing. Note that trianglesAOBandCOBare both isosceles.6ABC=6ABO+6OBC=12(180-6AOB)+12(180-6COB) =180-12(360-6AOC) =126AOC.Note that to make the proof rigorous, we also need to1
consider the cases of the angle being acute, right, and obtuse, separately, which essentiallyuses the same idea.What’s great about this is that if you imagine slidingBalong arcAC,6ABCwill beconstant.ABCDEHere, we’ll prove that6CEB=6AEDis equal to the average of arcsBCandAD.Proof:6AED=6CAB+6ACDby the exterior angle theorem. And6CAB+6ACD=12(6.0ptAD_+6.0ptBC_).