Question

true or false

1. the intersection of any two planes is a line. 1.3
2. some half - planes intersect. 1.4
3. some triangles are concave. 1.6
4. a polygon is both a simple curve and a closed curve. 1.6
5. no cones are polyhedra. 1.7
6. a prism is a cone and a cylinder. 1.7

Explanation:

Step1: Recall plane - intersection property

Two non - parallel planes intersect in a line. But if two planes are parallel, they do not intersect. So the statement "The intersection of any two planes is a line" is false.

Step2: Understand half - plane intersection

Half - planes can intersect. For example, if two half - planes are formed from intersecting planes, their intersection is non - empty. So the statement "Some half - planes intersect" is true.

Step3: Know triangle properties

All triangles are convex. By definition, a triangle is a three - sided polygon and it cannot be concave. So the statement "Some triangles are concave" is false.

Step4: Recall polygon definition

A polygon is a simple closed curve made up of line segments. So the statement "A polygon is both a simple curve and a closed curve" is true.

Step5: Understand polyhedron and cone

A polyhedron is a three - dimensional solid made up of polygons. A cone has a curved surface and is not made up of polygons. So the statement "No cones are polyhedra" is true.

Step6: Recall prism, cone and cylinder definitions

A prism is a polyhedron with two congruent and parallel bases connected by rectangular faces. A cone has a circular base and a vertex, and a cylinder has two circular bases. A prism is neither a cone nor a cylinder. So the statement "A prism is a cone and a cylinder" is false.

Answer:

1. False
2. True
3. False
4. True
5. True
6. False