Question

1. for each statement, determine if it is true or false. if false, explain why

a. two intersecting lines are coplanar. hint: think carefully about what it means for two lines to intersect.
b. three noncollinear points are always coplanar.
c. a line segment contains an infinite number of points.
d. the union of two rays is always a line.

Explanation:

Step1: Analyze statement a

Two intersecting lines share a common - point. By the definition of coplanar (lying in the same plane), two intersecting lines always lie in a single plane. So, it is true.

Step2: Analyze statement b

Three non - collinear points uniquely determine a plane. This is a fundamental postulate in geometry. So, three non - collinear points are always coplanar, and it is true.

Step3: Analyze statement c

A line segment is a part of a line. A line is made up of an infinite number of points, and a line segment, being a part of a line, also contains an infinite number of points. So, it is true.

Step4: Analyze statement d

The union of two rays is not always a line. If the two rays have a common endpoint and extend in opposite directions, their union is a line. But if they do not have the right orientation (e.g., if they form an angle), their union is not a line. So, it is false.

Answer:

a. True
b. True
c. True
d. False