Chapter 3.5, Problem 40HP

To determine

Determine whether the parallel lines are reflexive , symmetric , or transitive.Whether they form an equivalence relation.

Answer to Problem 40HP

Parallel lines forms an equivalence relation.

Explanation of Solution

Formula Used:

Corresponding Angle Theorem:

If two lines are parallel , then the corresponding angles are congruent.

Calculation : t

  

Let l, m , n be lines such that :

  $l\parallel m$ and $m\parallel n$

• Since coincidence lines are parallel , $l\parallel l$ . So, parallel lines are reflexive.
• If $l\parallel m$ , then clearly $m\parallel l$ . So, parallel lines are symmetric.
• Now , we prove that parallel lines are transitive:

Given: $l\parallel m$ and $m\parallel n$

To Prove: $l\parallel n$

Proof:

Statement	Reason
$\angle 1\cong \angle 2$	Corresponding Angle Theorem, since $l\parallel m$
$\angle 2\cong \angle 3$	Corresponding Angle Theorem, since $m\parallel n$
$\angle 1\cong \angle 3$	Transitive Property , since $\angle 1\cong \angle 2$ and $\angle 2\cong \angle 3$

And $\angle 1\text{ and }\angle 3$ are corresponding angles , where t is the transversal to the lines l and n .

By the Converse of Corresponding Angle Theorem which states that when a transversal intersects two lines , the two lines are parallel if the corresponding angles are congruent, the lines l and n are parallel.

So, $l\parallel n$ .

Hence , we can conclude that the parallel lines are reflexive, symmetric and transitive .

So, parallel lines form an equivalence relation.