Question

since \\( \angle coa \\) is complementary to \\( \angle aof, m \angle coa + m \angle aof = 90 ^ { \circ } \\). since \\( \angle eob \\) forms a vertical angle with \\( \angle aof \\), they are congruent by the vertical angle theorem. by the substitution property of equality, \\( m \angle coa + 40 ^ { \circ } = 90 ^ { \circ } \\). applying the subtraction property of equality gives \\( m \angle coa = 50 ^ { \circ } \\).
what is missing from the proof?
the proof did not explain why angle eob is congruent to angle aof \\( \downarrow \\) implies that \\( \downarrow \\). the proof should have used the
measure of angle bod equals measure of angle aof
measure of angle eob equals measure of angle aof
measure of angle eob equals measure of angle coa
measure of angle bod equals measure of angle coa

Explanation:

Step1: Recall vertical - angle property

Vertical angles are congruent. Since \(\angle EOB\) and \(\angle AOF\) are vertical angles, if we want to show that \(m\angle EOB=m\angle AOF\), we need to use the vertical - angle theorem. The proof already used the fact that \(\angle COA\) and \(\angle BOD\) are congruent (vertical angles). But for the relation between \(\angle EOB\) and \(\angle AOF\), the vertical - angle theorem (which states that if two angles are vertical angles, then they are congruent) is missing.

Step2: Analyze the given proof structure

The proof has a chain of reasoning: using the complementarity of \(\angle COA\) and \(\angle AOF\) (\(m\angle COA + m\angle AOF=90^{\circ}\)), and a wrong substitution (\(m\angle COA + 40^{\circ}=90^{\circ}\) instead of using the correct congruent angle relation). The key missing part is the application of the vertical - angle theorem for \(\angle EOB\) and \(\angle AOF\).

Answer:

measure of angle \(EOB\) equals measure of angle \(AOF\)