Fred and victoria provide the following proofs for vertical angles to be equal.

fred's proof: angle 2 + angle 3 = 180á (t is a straight line)
angle 1 + angle 2 = 180á (pq is a straight line)
therefore, angle 1 + angle 2 = angle 2 + angle 3 (transitive property of equality)
hence, angle 1 = angle 3 (subtraction property of equality)

victoria's proof: angle 1 + angle 4 = 180á (t is a straight line)
angle 1 + angle 2 =180á (pq is a straight line)
therefore, angle 1 + angle 2 = angle 1 + angle 4 (transitive property of equality)
hence, angle 2 = angle 4(subtraction property of equality

Both Fred and Victoria's proofs are correct.

Step-by-step explanation:

Fred's proof:

angle 2 + angle 3 = 180° (t is a straight line) IT IS TRUE THAT t IS A STRAIGHT LINEangle 1 + angle 2 = 180° (PQ is a straight line) IT IS TRUE THAT PQ IS A STRAIGHT LINETherefore, angle 1 + angle 2 = angle 2 + angle 3 (Transitive Property of Equality) THIS IS AN EXAMPLE OF THE TRANSITIVE PROPERTY OF EQUALITYHence, angle 1 = angle 3 (Subtraction Property of Equality)THIS IS AN EXAMPLE OF THE SUBTRACTION PROPERTY OF EQUALITY

Victoria's proof:

angle 1 + angle 4 = 180° (t is a straight line) TRUEangle 1 + angle 2 =180° (PQ is a straight line) TRUETherefore, angle 1 + angle 2 = angle 1 + angle 4 (Transitive Property of Equality) TRUEHence, angle 2 = angle 4 (Subtraction Property of Equality) TRU