Question

determine if each of the following statements are true or false. if false, explain why. a. two intersecting lines are coplanar. b. three noncollinear points are always coplanar. c. two planes can intersect in exactly one point. d. a line segment contains an infinite number of points. e. the union of two rays is always a line. a. choose the correct answer below. a. the statement is false because the two lines could be coplanar or they could be noncoplanar. b. the statement is false because the two lines could be skew lines and would be noncoplanar. c. the statement is true. d. the statement is false because the two lines could be parallel. b. choose the correct answer below. a. the statement is false because three noncollinear points are always noncoplanar. b. the statement is false because the three points can be coplanar but they could also be noncoplanar. c. the statement is true. d. the statement is false because the three points are noncollinear which means they do not lie on a single line. c. choose the correct answer below. a. the statement is false because they can intersect in a line, the empty set, or a plane. b. the statement is false because they can only intersect in a line.

Explanation:

Step1: Recall coplanar lines definition

Two intersecting lines always lie in the same plane, so they are coplanar.

Step2: Recall non - collinear points property

Three non - collinear points determine a unique plane, so they are always coplanar.

Step3: Recall plane - plane intersection

Two planes intersect in a line (when they are not parallel), not a single point.

Step4: Recall line segment property

A line segment is a part of a line and a line contains an infinite number of points, so a line segment also contains an infinite number of points.

Step5: Recall ray union property

The union of two rays is not always a line. For example, two rays with a common endpoint form an angle.

Answer:

a. C. The statement is true.
b. C. The statement is true.
c. B. The statement is false because they can only intersect in a line.
d. (No options provided, but the statement is true)
e. (No options provided, but the statement is false)