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 propert
what is vertical angles theorem
the definition of complementary
the definition of congruence
that linear pair theorem
to explain the missing statement.
\\( \

$$\begin{array} { l } { \\text { s congruent to angle } aof } \\\\ { \\text { re of angle } aof } \\\\ { \\text { implies } } \\end{array}$$

\\) the proof should have used

Explanation:

Step1: Use the property of vertical angles

Since \(\angle EOB\) and \(\angle AOF\) are vertical angles, by the vertical angles theorem, \(\angle EOB\cong\angle AOF\). Given \(m\angle EOB = 40^{\circ}\), so \(m\angle AOF=40^{\circ}\)

Step2: Use the definition of complementary angles

We know that \(\angle COA\) and \(\angle AOF\) are complementary (\(m\angle COA + m\angle AOF=90^{\circ}\)). Substitute \(m\angle AOF = 40^{\circ}\) into the equation: \(m\angle COA+40^{\circ}=90^{\circ}\)

Step3: Solve for \(m\angle COA\)

Using the subtraction property of equality, \(m\angle COA=90^{\circ}- 40^{\circ}\)

Answer:

\(m\angle COA = 50^{\circ}\)