Chapter 9 Theorems

Transversals Definition

A line that intersects two coplanar lines in two distinct points

Corollary to Triangle Sum Theorem 2

Acute angles in any right triangle are complementary

AIP Theorem

Given 2 lines cut by a transversal, if a pair of alternate interior angles are congruent, the lines are parallel

AEP Theorem

Given 2 lines cut by a transversal, if alternate exterior angles are congruent, the lines are parallel.

CAP Theorem

Given 2 lines cut by a transversal, if corresponding angles are congruent, then the lines are parallel.

SSEP

Given 2 lines cut by a transversal, if same side exterior angles are supplementary, then the lines are parallel.

SSIP Theorem

Given 2 lines cut by a transversal, if same side interior angles are supplementary, then the lines are parallel.

Corollary to Triangle Sum Theorem 1

If two angles of one triangle are congruent to two angles of a 2nd triangle, then the 3rd angles are also congruent.

PAI Theorem

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

PCA Theorem

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

PSSE Theorem

If two parallel lines are cut by a transversal, then same side exterior angles are supplementary.

PSSI Theorem

If two parallel lines are cut by a transversal, then same side interior angles are supplementary.

PAE Theorem

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

Theorem 9-10

In a plane, if a line intersects on of two parallel lines in only one point, then it intersects the other.

Theorem 9-12

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

Theorem 9-11

In a plane, if two planes are parallel to a third line, then they are parallel to each other.

Unnamed theorem 2

In a plane, two lines perpendicular to the same line are parallel.

Parallel Sets

Sets that are contained in parallel lines (lines, rays, segments)

Skew Sets

Sets that are contained in skew lines (lines, rays, segments) SKEW CAN ONLY BE APPLIED TO LINES OR PARTS OF LINES

Corollary to Triangle Sum Theorem 3

The Measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Triangle Sum Theorem

The sum of the measures of the angles of any triangle is 180 degrees.

Existence of Parallels Theorem

Through an external point, there exists at least 1 line parallel to the given line

Parallel Postulate

Through an external point, there is only 1 line parallel to a given line (proves uniqueness)

Parallel Lines

Two coplanar lines that do not intersect.

Skew Lines

Two non-coplanar lines that don't intersect

Unnamed theorem 1

Two parallel lines lie in exactly 1 plane.