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 statementreason 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 .
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