Relationships Between Lines Parallel Lines two lines that

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 P 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. l P * l

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. 1 2 <1 =~ <2 Theorem 3. 4 Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4 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. <3 ~ = <4 5 6 m<5 + m<6 = 180* 7 8 <7 ~ = <8 j h j _|_ k k

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

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

Proving Lines are Parallel j 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. 1 k 2 ~ If <1 = <2, Then j _|_ k 3 4 If <3 ~ = <4 Then j _|_ k 5 6 If m<5 + m<6 = 180* Then j _|_ k 7 8 ~ If <7 = <8 Then j _|_ k

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