Angle Chasing.pdf - Michael Sun Angle Chasing Angle chasing is an extremely important skill Here is a warm-up Triangles ABC and ADC are isosceles with

# Angle Chasing.pdf - Michael Sun Angle Chasing Angle chasing...

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April 10, 2016 Michael Sun Angle Chasing Angle chasing is an extremely important skill. Here is a warm-up: Triangles ABC and ADC are isosceles with AB = BC and AD = DC . Point D is inside 4 ABC , 6 ABC = 40 , and 6 ADC = 140 . What is the degree measure of 6 BAD ? Source: AMC 12, 2007 Solution: A C B D 6 ABC = 40 implies that 6 BAC = 6 BCA = 70 . 6 ADC = 140 implies that 6 DAC = 6 DCA = 20 . So 6 BAD = 70 - 20 = 50 . Hopefully these types of problems are pretty natural. In general, isosceles and equilateral triangles are great for angle chasing. Next, you should be familiar with the following from geometry: O B A C Prove that 6 ABC = 1 2 6 AOC . Proof: As hinted above, isosceles triangles are great for angle chasing. Note that triangles AOB and COB are both isosceles. 6 ABC = 6 ABO + 6 OBC = 1 2 (180 - 6 AOB )+ 1 2 (180 - 6 COB ) = 180 - 1 2 (360 - 6 AOC ) = 1 2 6 AOC. Note that to make the proof rigorous, we also need to 1 consider the cases of the angle being acute, right, and obtuse, separately, which essentially uses the same idea. What’s great about this is that if you imagine sliding B along arc AC , 6 ABC will be constant. A B C D E Here, we’ll prove that 6 CEB = 6 AED is equal to the average of arcs BC and AD . Proof: 6 AED = 6 CAB + 6 ACD by the exterior angle theorem. And 6 CAB + 6 ACD = 1 2 (6.0ptAD _ + 6.0ptBC _ ) .  • • • 