Question

extra practice
in exercises 1–4, use the diagram.

1. name three points.
2. name two lines.
3. name all points in plane h.
4. name the plane that contains points a, b, and g.

in exercises 5–8, use the diagram.

1. name one pair of opposite rays.
2. name two points that are collinear with point d.
3. name the point of intersection of line cd with plane a.
4. name a point that is not coplanar with plane a.

in exercises 9–11, sketch the figure described.

1. plane a and line $overline{bc}$ intersecting at point c
2. plane m and plane n not intersecting
3. lines a, b, and c intersecting at three points
4. a tripod can be used to level a camera. what geometric figure is modeled by the intersection of a tripod to the ground? explain.

in exercises 13 and 14, graph the inequality on a number line. tell whether the graph is a segment, a ray or rays, a point, or a line.

1. $x \geq 2$
2. $-4 < x < 4$
3. what is the greatest number of intersection points that two planes can have? what is the fewest number of intersection points that two planes can have?

Explanation:

Question 15:

Step1: Recall the properties of planes

Two planes either are parallel (and thus have no intersection points) or intersect in a line (which contains infinitely many points).

Step2: Determine the number of intersection points

• For the greatest number of intersection points: When two planes intersect, their intersection is a line. A line has infinitely many points, so the greatest number of intersection points is infinitely many.
• For the fewest number of intersection points: If two planes are parallel, they do not intersect at all, so the fewest number of intersection points is 0. If they are not parallel, they intersect in a line (infinitely many points). So the minimum number of intersection points (when considering non - parallel planes, the intersection is a line with infinite points; when parallel, 0) but if we consider the case of non - parallel planes, the intersection is a line (infinite points), and for parallel planes, 0. But the question is about the greatest and fewest. The greatest is infinite (since a line has infinitely many points), the fewest is 0 (when planes are parallel, no intersection points). Wait, but actually, two planes either are parallel (0 intersection points) or intersect in a line (infinitely many intersection points). So:

Greatest number of intersection points: Infinitely many (because the intersection of two non - parallel planes is a line, which has infinitely many points).

Fewest number of intersection points: 0 (when the two planes are parallel, they do not intersect at all).

Answer:

The greatest number of intersection points that two planes can have is infinitely many (because two non - parallel planes intersect in a line, which contains infinitely many points). The fewest number of intersection points that two planes can have is 0 (when the two planes are parallel, they do not intersect).