Q: Do all coplanar segments intersect

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No. If two lines intersect, then they're definitely coplanar.

a triangle

Angles that are separated by a distance. For example, any two angles of any polygon do not intersect and they are coplanar.

A polygon, but only if all the line segments are coplanar.

Parallel lines.

Related questions

If rays, parallel ones. If segments there are many more options because you can have a segment that is not long enough to reach the point of intersection.

Coplanar lines that do not intersect are called parallel lines.

are two lines that are not parallel, coplanar, and do not intersect

No. If two lines intersect, then they're definitely coplanar.

a triangle

Coplanar lines can intersect an infinite amount of times.

Coplanar lines that do not intersect are parallel. Non-coplanar lines that do not intersect are called skew lines.

Angles that are separated by a distance. For example, any two angles of any polygon do not intersect and they are coplanar.

parallel

Two coplanar lines that never intersect are called called parallel lines.

A polygon, but only if all the line segments are coplanar.

parallel lines