Lines AB and CD meet at point O. Angle AOC is 58. Find the angle BOD.

From the condition it is known that lines AB and CD meet at point O. It is also known that the angle AOC is 58 °. We need to find the degree measure of the BOD angle.
So, if we draw a drawing, we can see that the AOC and BOD angles are vertical angles.
Let’s remember the definitions of vertical angles.
Vertical corners are pairs of corners with a common vertex that are formed when two straight lines intersect so that the sides of one corner are a continuation of the sides of the other.
When two straight lines intersect, we have two pairs of vertical angles and they are equal to each other.
Hence the angle BOD = angle AOC = 58 °.