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 S′ represent the degreesum of angles BAD and ABC. If r=S/S′, 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=∠CDE+∠DCE=180∘−α+180∘−β
=360∘−(α+β)
S′=∠BAD+∠ABC=360−(α+β)∴r=S/S′=1
Method II. \mathrm{S}=\angle \mathrm{CDE}+\angle \mathrm{DCE}=180^{\circ}-\angle \mathrm
S′=∠BAD+∠ABC=180∘−∠E∴r=S/S′=1
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