Problem: If ΔA1​A2​A3​ is equilateral and An+3​ is the midpoint of line segment An​An+1​ for all positive integers n, then the measure of X44​A45​A43​ equals
Answer Choices:
A. 30∘
B. 45∘
C. 60∘
D. 90∘
E. 120∘
Solution:
Triangle A2​A3​A4​ has vertex angles 60∘,30∘. 90∘, respectively. Since ∠A1​A2​A3​=60∘, and A2​A4​ and A2​A5​ have the same length, ΔA2​A4​A5​ is equilateral. Therefore, ΔA3​A4​A5​ has vertex angles 30∘,30∘,120∘. respectively. Then ΔA4​A5​A6​ has vertex angles 30∘,60∘,90∘, respectively.
Finally, since A4​A5​A6​=60∘ and A5​A6​ and A5​A7​ have the same length, △A5​A6​A7​ is again equilateral. Therefore △An​An+1​An+2​ ∼ΔAn+4​An+5​An+6​ with An​ and An+4​ as corresponding vertices. Thus ∠A44​A45​A43​=∠A4​A5​A3​=120∘.
