Given: ∆ABC with AB = c, BC = a, AC = b Prove: = = Statement Reason 1. Draw a line through B that is perpendicular to and label the point of intersection with as D. construction 2. In ∆ABD, BD = c sin A definition of sine in right triangle 3. In ∆CBD, BD = a sin C definition of sine in right triangle 4. c sin A = a sin C Substitution Property of Equality 5. = dividing throughout by sin A sin C 6. Draw a line through A that is perpendicular to and label the point of intersection with as E. construction 7. In ∆BAE, AE = c sin B 8. In ∆CAE, AE = b sin C 9. c sin B = b sin C Substitution Property of Equality 10. = dividing throughout by sin B sin C 11. = = Transitive Property of Equality A. definition of cosine in right triangle B. definition of sine in right triangle C. definition of tangent in right triangle D. Corresponding sides of congruent triangles are equal. E. definition of equilateral triangle