Parallel Lines and Planes Chapter 3

Section 3-3

Theorem If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel. Which angles must be congruent to prove lines r and s are parallel?

Postulate If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. Which angles need to be congruent in order for lines r and s to be parallel?

Theorem If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. If <4 is congruent to <6 (or <3 is congruent to <5), then lines r and s are parallel.

Theorem If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel. Which angles must be supplementary to prove lines r and s are parallel?

Ways to Show Lines are Parallel Show a pair of corresponding angles are congruent Show a pair of alternate interior angles are congruent Show a pair of same-side interior angles are supplementary In a plane, show that both lines are perpendicular to a third line Show that both lines are parallel to a third line

Homework- Proving Lines Parallel 3-3 Practice Worksheet (7-17 all)

Proof PPT

Section 3-4 p.164

Theorem If two parallel planes are cut by a third plane, then the lines of intersection are parallel Examples: Floor and ceiling intersected by a wall

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

Theorem In a plane, two lines perpendicular to the same line are parallel. If line t is perpendicular to line k and line t is also perpendicular to line l, then lines l and k are parallel.

If two lines are parallel to the same line, then they are parallel to each other.

Section 3-5 p.171

More Theorems Through a point not on a line, there is exactly one line parallel to the given line. Through a point not on a line, there is exactly one line perpendicular to the given line.

Triangle Angle Sum Theorem The sum of the measures of the angles of a triangle is 180.

Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Ratio of Angles in a Triangle The ratio of the angles in a triangle is 2:3:4 What are the measures of the three angles? 2x + 3x + 4x = 180 x = 20 40°, 60°, and 80°

Homework p.167-169 #2, 7 p.175-176 #9, 12-14, 17, 20, 22-24