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

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 *

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

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?

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

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 ~ ~ ~ ~

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 = Use the Properties of Parallel Lines to Solve for x 125* 4 (x + 15*)

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 ~

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 ~ ~