Question

what is the reason for step 3 of this proof?
a. alternate interior angles theorem
b. corresponding angles in congruent triangles are congruent.
c. for parallel lines cut by a transversal, corresponding angles are congruent.
d. vertical angles theorem
e. sas criterion for congruence
reset next
prove: quadrilateral abcd is a parallelogram.
proof
statement
reason
\\( \overline { a c } \\) and \\( \overline { b d } \\) bisect each other.
given
\\( \begin{array} { l } { a e = e c } \\ { b e = e d } end{array} \\)
definition of bisection
\\( m \angle a e b = m \angle c e d \\)
\\( \triangle a b e \cong \triangle c d e \\)
sas criterion
\\( \angle a c d \cong \angle c a b \\)
corresponding angles of congruent
triangles are congruent.
converse of alternate interior angles
theorem
\\( m \angle b e c = m \angle a e d \\)
vertical angles theorem
\\( \triangle b e c \triangle d e a \\)
sas criterion for congruence
\\( \angle d b c \cong \angle b d a \\)
corresponding angles of congruent
triangles are congruent.
converse of alternate interior angles
theorem

1. quadrilateral abcd is a parallelogram

definition of a parallelogram

Explanation:

Vertical angles are the angles opposite each other when two lines cross. In the given proof, \( \angle AEB \) and \( \angle CED \) are vertical angles. The Vertical Angles Theorem states that vertical angles are congruent, which means \( m\angle AEB=m\angle CED \).

Answer:

D. Vertical Angles Theorem