UpStudy Free Solution:

To complete the proof, let's fill in each blank with the appropriate theorem or property.

Given: \(m \angle AOB = 42^ \circ \) and \(m \angle EOF = 66^ \circ \).

By the \(\textbf { Vertical Angle Theorem} \), \(\angle EOF \cong \angle BOC\).

Therefore, \(m \angle BOC = 66^ \circ \).

By the \(\textbf { Angle Addition Postulate} \), \(m \angle AOC = m \angle AOB + m \angle BOC = 42^ \circ + 66^ \circ = 108^ \circ \).

By the \(\textbf { Linear Pair Postulate} \), \(m \angle AOC + m \angle COD = 180^ \circ \).

After application of the Subtraction Property of Equality, \(m \angle COD = 180^ \circ - 108^ \circ = 72^ \circ \).

Summary of the Completed Proof

It is given that \(m \angle AOB = 42^ \circ \) and \(m \angle EOF = 66^ \circ \).

By the Vertical Angle Theorem, \(\angle EOF \cong \angle BOC\).

Therefore, \(m \angle BOC = 66^ \circ \).

By the Angle Addition Postulate, \(m \angle AOC = 108^ \circ \).

By the Linear Pair Postulate, \(m \angle AOC + m \angle COD = 180^ \circ \).

After application of the Subtraction Property of Equality, \(m \angle COD = 72^ \circ \).

Thus, the correct answers for each drop-down menu are:

Vertical Angle Theorem

Angle Addition Postulate

Linear Pair Postulate

Subtraction Property of Equality

Supplemental Knowledge

In geometry, understanding the relationships between angles is crucial for solving many problems. Here are some key concepts that relate to the problem at hand:

Vertical Angles Theorem: When two lines intersect, they form two pairs of opposite (vertical) angles that are congruent. For example, if two lines intersect at point O, forming angles \(\angle AOB\) and \(\angle EOF\), then \(\angle AOB \cong \angle EOF\).

Linear Pair Postulate: If two angles form a linear pair (they are adjacent and their non-common sides form a straight line), then their measures add up to 180 degrees. For instance, if \(\angle AOC\) and \(\angle COD\) form a linear pair, then \(m\angle AOC + m\angle COD = 180^ \circ \).

Angle Addition Postulate: If a point lies in the interior of an angle, the measure of the larger angle is the sum of the measures of the two smaller angles formed.

These principles help in understanding how different angles relate to each other when lines intersect or when dealing with adjacent angles.

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