Chapter 3.5, Problem 42HP

To determine

Summarize the five methods used to prove that the two lines are parallel.

Explanation of Solution

Calculation : s t

  

Let l, m be lines , t is transversal such that :

  $s\perp m$ and $s\perp l$

  $\angle 1\cong \angle 8$

  $m\angle 3+m\angle 5=180°$

  $\angle 4\cong \angle 5$

  $\angle 2\cong \angle 6,\angle 4\cong \angle 8,\angle 1\cong \angle 5,\angle 3\cong \angle 7$

• Alternate Exterior Angle Converse :

If two lines in a plane are cut by a transversal such that a pair of alternate exterior angles is congruent , then the lines are parallel.

t

  

Since , $\angle 1\cong \angle 8$ and is a pair of alternate exterior angles , so, $l\parallel m$

• Consecutive Interior Angles Converse :

If two lines in a plane are cut by a transversal such that a pair of consecutive interior angles is supplementary , then the lines are parallel.

t

  

Since , $m\angle 3+m\angle 5=180°$ , so , $\angle 3\text{ and }\angle 5$ are supplementary and is a pair of consecutive interior angles , so $l\parallel m$

• Alternate Interior Angle Converse :

If two lines in a plane are cut by a transversal such that a pair of alternate interior angles is congruent , then the lines are parallel.

t

  

Since , $\angle 4\cong \angle 5$ and is a pair of alternate interior angles . So, $l\parallel m$

• Converse of Corresponding angles Postulate :

If two lines in a plane are cut by a transversal such that corresponding angles are congruent , then they are parallel.

t

  

Since , $\angle 2\cong \angle 6,\angle 4\cong \angle 8,\angle 1\cong \angle 5,\angle 3\cong \angle 7$ and are pairs of corresponding angles . So, $l\parallel m$

• Perpendicular Transversal Converse :

In a plane , if two lines are perpendicular to the same line, then the two lines are parallel.

  

Since , $l\perp s$ and $m\perp s$ , so, $l\parallel m$