Perpendicularity and Parallel Lines

Perpendicularity

A line and a plane are perpendicular if they intersect and if every line lying in the plane and passing through the point of intersection is perpendicular to the given line.

Basic Theorem of Perpendicular

If a line is perpendicular to each of 2 lines at their point of intersection, then it is perpendicular to the plane that contains them.

Through a given point of a given line...

there passes a plane perpendicular to the given line.

If a line and a plane are perpendicular...

then the plane contains every line perpendicular to the given line at its point of intersection with the given plane.

Through a given point of a given line..

there is only one plane perpendicular to the line

Perpendicular Bisecting Plane Theorem

The perpendicular bisecting plane of a segment is the set of all points equidistant from the endpoints of the segment.

Two lines perpendicular to the same plane are...

coplanar

Through a given point, there passes one and only one plane

perpendicular to a given line

Through a given point, there passes one and only one line

perpendicular to a given plane

Second Minimum Theorem

The shortest segment to a plane from an external point is the perpendicular segment. The distance to a plane from an external point is the length of the perpendicular segment from the point to the plane.

The distance to a plane from an external point is...

the length of the perpendicular segment from the point to the plane.

If two lines are cut by a transversal, and one pair of alternate interior angles are congruent...

then the other pair of alternate interior angles are also congruent.

If two lines are cut by a transversal so that a pair of alternate interior angles are congruent...

then the lines are parallel.

If two lines are cut by a transversal so that a pair of interior angles on the same side of the transversal are supplementary...

then the lines are parallel.

If two lines are cut by a transversal so that a pair of alternate exterior angles are congruent

then the lines are parallel.

In a plane, if two lines are both perpendicular to the same line

then they are parallel

If two parallel lines are cut by a transversal (alt int)

then alternate interior angles are congruent.

If two parallel lines are cut by a transversal (corresp)

then each pair of corresponding angles are congruent.

If two parallel lines are cut by a transversal (int)

then the interior angles on the same side of the transversal are supplementary

If two parallel lines are cut by a transversal... (alt ext)

then alternate exterior angles are congruent.

In a plane, if two lines are each parallel to a third line

then they are parallel to each other.

In a plane, if a line is perpendicular to one of two parallel lines...

then it is perpendicular to the other.