Question

1. sketch plane m intersecting plane n. then sketch plane o so that it intersects plane n, but not plane m.

Explanation:

Step 1: Intersect Planes \( M \) and \( N \)

Two planes in 3D space intersect in a unique line (by the postulate of plane intersection). So, draw two parallelograms (representing planes \( M \) and \( N \)) that overlap along a straight line (their line of intersection).

Step 2: Draw Plane \( O \) with Required Intersections

To have plane \( O \) intersect \( N \) (so they share a line) but not \( M \), we can position \( O \) such that its intersection line with \( N \) is parallel to the intersection line of \( M \) and \( N \), and \( O \) does not extend to meet \( M \). Geometrically, non - intersecting planes (like \( M \) and \( O \)) can be parallel or skew - related, but since both intersect \( N \), making their intersection lines with \( N \) parallel ensures \( M \) and \( O \) do not intersect.

Answer:

(Visual Sketch Description)

1. Sketch Plane \( M \) and Plane \( N \) Intersection:

• Draw two intersecting planes, \( M \) and \( N \). Represent each plane as a parallelogram (a common way to sketch planes in 3D). Let their intersection be a line (since two planes intersect in a line). Label the line of intersection, say \( l \), where \( M \cap N = l \).

1. Sketch Plane \( O \):

• Draw plane \( O \) as another parallelogram. Ensure that plane \( O \) intersects plane \( N \) (so they share a line of intersection, say \( m \), where \( O \cap N = m \)) and that plane \( O \) does not intersect plane \( M \). This means plane \( O \) should be parallel to the line of intersection of \( M \) and \( N \) (or positioned such that it does not meet \( M \) at all; geometrically, if \( M \) and \( N \) intersect along \( l \), plane \( O \) can be drawn such that it intersects \( N \) along a line \( m \) that is parallel to \( l \) and does not intersect \( M \)).

(Note: Since this is a sketching problem, the key is to represent the three planes with the correct intersection properties. A rough 3D sketch with labeled planes \( M \), \( N \), \( O \), showing \( M \cap N \) as a line, \( O \cap N \) as a line, and no intersection between \( M \) and \( O \) would satisfy the problem.)