To determine the measures of the angles AOD, BOD, and COB, we need to analyze the given figure and apply the relevant concepts of angles formed by intersecting lines. Let's break down the solution step by step:

Step 1: Understand the problem and figure
- We are given a figure with two straight lines AB and CD.
- The lines intersect at a point, which is denoted as "O."

Step 2: Identify relevant angles and relationships
- AOC is given as 42°.
- The angles AOC and COB are called vertical angles and are equal in measure.
- The angles AOD and BOD are linear pairs and their measures add up to 180°.

Step 3: Calculate the measures of the angles
- Since AOC and COB are vertical angles, their measures are equal. Therefore, the measure of COB is also 42°.
- Since AOD and BOD are linear pairs, their measures add up to 180°. Therefore, to find the measure of AOD or BOD, we subtract the measure of COB (42°) from 180°:
- Measure of AOD = 180° - 42° = 138°
- Measure of BOD = 180° - 42° = 138°

Step 4: Summarize the answers
The measures of the angles are as follows:
- AOD = 138°
- BOD = 138°
- COB = 42°

Step 5: Explain the reasoning and relationships
- The angle AOC is given as 42°, and since angles AOC and COB are vertical angles, they have equal measures.
- The angles AOD and BOD are linear pairs, which means they are adjacent angles formed by intersecting lines and their measures add up to 180°.

Step 6: Provide a visually appealing response
The measures of the angles in the given figure are:
- AOD = 138°
- BOD = 138°
- COB = 42°

Note: Remember to format the response in a visually appealing manner, using appropriate HTML tags for headings and bullet points.