Question

given: \\( \overline { b c } \\) is parallel to \\( \overline { e d } \\)
\\( m \angle a b c = 70 ^ { \circ } \\)
\\( m \angle c e d = 30 ^ { \circ } \\)
prove: \\( m \angle b e c = 40 ^ { \circ } \\)
which of the following accurately completes the missing statement and
justification of the two - column proof?
\\( m \angle b e c + 30 ^ { \circ } = 70 ^ { \circ } \\); substitution property of equality
\\( m \angle b e c + 30 ^ { \circ } = 70 ^ { \circ } \\); addition property of equality
\\( m \angle b e c + 40 ^ { \circ } = 70 ^ { \circ } \\); substitution property of equality
\\( m \angle b e c + 40 ^ { \circ } = 70 ^ { \circ } \\)

Explanation:

Step1: Substitute known angles

We know that \(m\angle BEC + m\angle CED=m\angle BED\) (Angle Addition Postulate). Given \(m\angle CED = 30^{\circ}\) and \(m\angle BED=m\angle ABC = 70^{\circ}\) (Corresponding Angles Theorem). Substitute \(m\angle CED\) and \(m\angle BED\) into the equation \(m\angle BEC + m\angle CED=m\angle BED\).
So, \(m\angle BEC+30^{\circ}=70^{\circ}\). The property used for substitution is the Substitution Property of Equality.

Answer:

\(m\angle BEC + 30^{\circ}=70^{\circ}\); Substitution Property of Equality