1. Relationships Between Lines Parallel Lines – two lines that are coplanar and do not intersect Skew Lines – two lines that are NOT coplanar and do not intersect Perpendicular Lines – two lines that intersect to form a right angle Parallel Planes – two planes that do not intersect

2. Parallel and Perpendicular Postulates Postulate 13: Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Postulate 14: Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given point. P*P* l l P *

3. Transversals A transversal is a line that intersects two or more coplanar lines at different points.

4. Angles Formed By Transversals Corresponding Angles – two angles are corresponding if they occupy corresponding positions as a result of the intersection of the lines. (1 & 5) Alternate Exterior Angles – two angles are alternate exterior if they lie outside the two lines on opposite sides of the transversal (1 & 7) Alternate Interior Angles – Two angles that lie between the two lines on opposite sides of the transversal (4 & 6) Consecutive Interior Angles – two angles that lie between the two lines on the same side of the transversal (4 & 5) Who can name another pair for each of the angle types above?

5. Postulates and Theorems of Parallel Lines Postulate 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Theorem 3.4 Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Theorem 3.5 Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Theorem 3.5 Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem 3.7 Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. 1 2 <1 = <2 ~ 3 4 <3 = <4 ~ 7 8<7 = <8 ~ 5 6 m<5 + m<6 = 180* j h k j _|_ k

6. Proving the Alternate Interior Angles Theorem Given: p || q Prove: <1 = <2 StatementsReasons 1.p || qGiven 2.<1 = <3Corresponding Angles Postulate 3.<3 = <2Vertical Angles Theorem 4.<1 = <2Transitive Property of Congruence 2 1 p 3 q ~ ~ ~ ~

7. Using Properties of Parallel Lines Given that m<5 = 65*, find each measure. Tell which postulate or theorem you use. a.m<6 = b.m<7 = c.m<8 = d.m<9 = 5 9 8 7 6 Use the Properties of Parallel Lines to Solve for x 125* 4 (x + 15*)

8. Proving Lines are Parallel Postulate 16: Corresponding Angle Converse: If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. Theorem 3.8: Alternate Interior Angles Converse: If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel. Theorem 3.9 Consecutive Interior Angles Converse: If two lines are cut by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. Theorem 3.10 Alternate Exterior Angles Converse - If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. j 1 2 If <1 = <2, Then j _|_ k ~ k 3 4 If <3 = <4 Then j _|_ k ~ 5 6 If m<5 + m<6 = 180* Then j _|_ k 7 8If <7 = <8 Then j _|_ k ~

9. Proving the Alternate Interior Angles Converse Given: <1 = <2 Prove: p _|_ q StatementsReasons 1.<1 = < 2Given 2.<2 = <3 Vertical Angles Theorem 3.<1 = < 3Transitive Property of Congruence 4.p _|_ qCorresponding Angles Converse 1 2 p q ~ ~ 3 ~ ~