Problem: Let side AD of convex quadrilateral ABCD be extended through D and let side BC be extended through C, to meet in point E. Let S represent the degree-sum of angles CDE and DCE and let Sβ² represent the degreesum of angles BAD and ABC. If r=S/Sβ², then:
Answer Choices:
A. r=1 sometimes, r>1 sometimes
B. r=1 sometimes, r<1 sometimes
C. 0<r<1
D. r>1
E. r=1
Solution:
Method I. S=β CDE+β DCE=180ββΞ±+180ββΞ²
=360ββ(Ξ±+Ξ²)
Sβ²=β BAD+β ABC=360β(Ξ±+Ξ²)β΄r=S/Sβ²=1
Method II. \mathrm{S}=\angle \mathrm{CDE}+\angle \mathrm{DCE}=180^{\circ}-\angle \mathrm
Sβ²=β BAD+β ABC=180βββ Eβ΄r=S/Sβ²=1
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