Proofs involving CPCTC

CPCTC

The main reason for proving two triangles congruent is to prove parts of those triangles are congruent.  Recall that a triangle congruence such as  means six things.  Three pairs of sides are congruent and three pairs of angles are congruent:

,

Study the following problems carefully so you understand the details.  Proofs are given in statement-reason format and in paragraph format.

Problem 1

Given:    and  is the midpoint of

Prove:   is the midpoint of

Note:  The symbol  means "is parallel to."

Proof:

 Statement Reason Given Alternate interior angles are congruent is the midpoint of Given Definition of "midpoint" Vertical angles are congruent ASA CPCTC is the midpoint of Definition of "midpoint"

In paragraph form, we could write the proof like this:

It is given that .  Therefore  since these are alternate interior angles for transversal .  Since  is the midpoint of , .  We also know  since these are vertical angles.  Therefore  by ASA, and so  since these are corresponding parts of the congruent triangles.  Thus  is the midpoint of .

Problem 2

Given:   , and

Prove:

Proof:

 Statement Reason Given Given Given SAS CPCTC If corresponding angles are congruent then the lines are parallel.

In paragraph form we can write:

Since , , and , it follows that  by SAS.  Therefore  since these are corresponding parts of the congruent triangles.  These are corresponding angles for  and  with transversal , so .

Problem 3

Given:   is the midpoint of , and

Prove:

Proof:

 Statement Reason is the midpoint of Given Definition of "midpoint" Given Corresponding angles of parallel lines are congruent. Given SAS CPCTC Given (we use this given a second time) Corresponding angles of parallel lines are congruent. Transitive property of congruence If corresponding angles are congruent the lines are parallel.

This proof is tricky.  Paragraph form might make it clearer:

Since  is the midpoint of .  Since ,  since these are corresponding angles for the parallel lines with transversal .  We also know , so  by SAS.  Therefore  since these are corresponding parts of the congruent triangles.  Now if we consider transversal  for parallel lines  and ,  since these are corresponding angles.  Now since  and , it follows that  by the transitive property.  But these are corresponding angles for lines  and  with transversal , so .