E A. Shiryayeva Problem Book (OGE 2012) 19. Analysis of geometric statements. Part 1. FIPI Task. Indicate (circle

ОкеанThe problem you have given is from the "E. A. Shiryayeva Problem Book (OGE 2012)" and it involves analyzing geometric statements. In this problem, you are required to indicate (circle) the numbers of the correct statements related to initial geometric knowledge such as segments, lines, and angles. Let"s analyze each statement one by one and determine whether it is correct or not.

1. A point lying on the perpendicular bisector of a segment is equidistant from the ends of that segment.

This statement is correct. When a point lies on the perpendicular bisector of a segment, it is equidistant from the endpoints of that segment. This is a fundamental property of the perpendicular bisector.

2. There exist three lines that pass through a single point.

This statement is incorrect. Only one line can pass through a single point. If three lines were to pass through a single point, they would coincide and form a single line.

3. Adjacent angles are always equal.

This statement is incorrect. Adjacent angles are angles that share a common ray and a common vertex, but they do not have to be equal. It is possible for them to have different measures.

4. Vertical angles are equal.

This statement is correct. Vertical angles are formed by the intersection of two lines and are opposite to each other. They are always equal in measure.

5. One of the two adjacent angles is always acute, while the other is obtuse.

This statement is incorrect. Adjacent angles can both be acute, both be obtuse, or one acute and the other obtuse. There is no specific relationship between their measures.

6. Only one line can be drawn through a given point in a plane.

This statement is incorrect. Through a single point in a plane, infinitely many lines can be drawn. It is a basic geometric property that a line can be determined by two points.

7. *Text is cut off*

Unfortunately, the text of statement #7 is not visible. Please provide the full text of statement #7 so that I can analyze it.

In summary, the correct statements among the given options are:

1. A point lying on the perpendicular bisector of a segment is equidistant from the ends of that segment.
4. Vertical angles are equal.

I hope this explanation has helped you understand the reasoning behind the correct statements.