QUESTION 2 [Total marks: 16] Consider the following statement: Let A,B,C,D be four distinct points no three of which are collinear such that A and D are on opposite sides of BC. Then if|ZABC| = |ZBCD], AB is parallel to CD. A line-by-line proof of this statement is given below. For each of the numbered lines (1)-(ix) below, give an explanation for the validity of the conclusions of that line. Quote relevant definitions, axioms and theorems where appropriate. In addition, sketch a suitable diagram to accompany the proof. () We consider these two cases: In Case 1, AB and CD do not meet and the result is proven (i) In Case 2, AB and CD do meet at point E. Assume that D lies between C and E. (II) Construct the point F on AB, with |BF| = |CE| and with F, E on opposite sides of BC (iv) ACBF ABCE. () ZFCB1 = \ZEBC. (vi) ZFCE = 1ZFCB+ ZBCE (vii) = 1ZFBE (vii) (x) Hence CD = AB and so these lines are parallel = 180 TC Market QUESTION 2 [Total marks: 16] Consider the following statement: Let A,B,C,D be four distinct points no three of which are collinear such that A and D are on opposite sides of BC. Then if|ZABC| = |ZBCD], AB is parallel to CD. A line-by-line proof of this statement is given below. For each of the numbered lines (1)-(ix) below, give an explanation for the validity of the conclusions of that line. Quote relevant definitions, axioms and theorems where appropriate. In addition, sketch a suitable diagram to accompany the proof. () We consider these two cases: In Case 1, AB and CD do not meet and the result is proven (i) In Case 2, AB and CD do meet at point E. Assume that D lies between C and E. (II) Construct the point F on AB, with |BF| = |CE| and with F, E on opposite sides of BC (iv) ACBF ABCE. () ZFCB1 = \ZEBC. (vi) ZFCE = 1ZFCB+ ZBCE (vii) = 1ZFBE (vii) (x) Hence CD = AB and so these lines are parallel = 180 TC Market