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 PostulateIf 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

*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~

34

<3 = <4~

7

8 <7 = <8~

5

6

m<5 + m<6 = 180*

j h

kj _|_ k

Proving the Alternate Interior Angles Theorem

Given: p || qProve: <1 = <2

Statements Reasons1. p || q Given2. <1 = <3 Corresponding Angles Postulate3. <3 = <2 Vertical Angles Theorem4. <1 = <2 Transitive Property of Congruence

21 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 =

59

876

Use the Properties of Parallel Lines to Solve for x

125*

4(x + 15*)

Proving Lines are ParallelPostulate 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

34

If <3 = <4Then j _|_ k

~

5

6

If m<5 + m<6 = 180*Then j _|_ k

7

8 If <7 = <8Then j _|_ k

~

Proving the Alternate Interior Angles Converse

Given: <1 = <2Prove: p _|_ q

Statements Reasons1. <1 = < 2 Given2. <2 = <3 Vertical Angles Theorem3. <1 = < 3 Transitive Property of Congruence4. p _|_ q Corresponding Angles Converse

12 p

q

~~

3~

~