Chapter 3 Notes

Section 3.1 Parallel Lines and Transversals

Parallel lines—are coplanar and do not intersect

 

 

 

Perpendicular lines—are coplanar and intersect to form four 90 degree angles.

 

 

 

Skew lines—are not coplanar and do not intersect

 

 

 

 

Parallel planes—do not intersect 

 

 

 

 

 

 

Transversal—a line that intersects two or more coplanar lines. (line t)

 

 

 

Corresponding angles—angles in corresponding positions

 

 

 

Alternate Exterior angles—lie outside the two lines (a and b) on opposite sides of the transversal (line t) 

 

 

 

Alternate Interior angles—lie between the two lines (a and b) on opposite sides of the transversal (line t)

 

 

Consecutive Interior angles—lie between the two lines (a and b) on the same side of the transversal (line t)   These are also know as same-side interior angles

 

 

 

Section 3.2 Angles and Parallel Lines

Postulate 3.1: (The Corresponding Angles Postulate) 

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

 

 

 

Theorem 3.1:  (The Alternate Interior Angles Theorem)

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

 

 

 

Theorem 3.2:  (The Consecutive Interior Angles Theorem)

If two parallel lines are cut by a transversal,
then Consecutive Interior angles are supplementary.

 

 

 

Theorem 3.3: (The Alternate Exterior Angles Theorem)

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

 

 

 

Theorem 3.4: (The Perpendicular Transversal Theorem)
 If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

 

 

 

 

An angled transversal???

 

 

 

Triangle Sum Theorem:

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

 

 

 

 

 

Section 3.5 Proving Lines Parallel

Postulate 3.4:  (The Corresponding Angles Converse)

If lines are cut by a transversal so that a pair of
Corresponding Angles
are congruent
,
then the lines are parallel.

 

 

 

 

Theorem 3.5: (The Alternate Exterior Angles Converse)

 If lines are cut by a transversal so that a pair of
Alternate Exterior Angles
are congruent
,
then the lines are parallel.
 

 

 

 

Theorem 3.6: (The Consecutive Interior Angles Converse)

If lines are cut by a transversal so that a pair of
Consecutive Interior Angles
are supplementary
,
then the lines are parallel. 

 

 

 

Theorem 3.7: (The Alternate Interior Angles Converse)

If lines are cut by a transversal so that a pair of
Alternate Interior Angles
are congruent
,
then the lines are parallel. 

 

 

 

 

Theorem 3.8: (The two perpendiculars are parallel theorem)

If two lines are perpendicular to the same third line,  
then the two lines are parallel. 

 

 

 

 

Section 3.6 Perpendiculars and Distance

The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point. 

 

 

The distance between two parallel lines is the length of the segment from a point on one line to the other line in a perpendicular direction.

 

 

 

In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.