Using Congruent Triangles :
Knowing that all pairs of corresponding parts of congruent triangles are congruent can help us to reach conclusions about congruent figures.
For instance, suppose we want to prove that
∠PQS ≅ ∠RQS
in the diagram shown below. One way to do this is to show that ΔPQS ≅ ΔRQS by the SSS congruence postulate. Then we can use the fact that corresponding parts of congruent triangles are congruent to conclude that
∠PQS ≅ ∠RQS
Problem 1 :
In the diagram shown below, AB  CD, BC  DA. Prove that
AB ≅ CD
Plane for Proof :
Show that ΔABD ≅ ΔCDB. Then use the fact that corresponding parts of congruent triangles are congruent.
Solution :
First copy the diagram and mark it with the given information. Then mark any additional information that we can deduce. Because AB and CD are parallel segments intersected by a transversal, BC and DA are parallel segments intersected by a transversal, we can deduce that two pairs of alternate interior angles are congruent.
Mark given information :
Add deduced information :
Paragraph Proof :
Because AB  CD, it follows from the Alternate Interior Angles Theorem, that ∠ABD ≅ ∠CDB. For the same reason, ∠ADB ≅ ∠CBD, because BC  DA. By the Reflexive Property of Congruence, BD ≅ BD. We can use the ASA congruence postulate to conclude that
ΔABD ≅ ΔCDB.
Finally, because corresponding parts of congruent triangles are congruent, it follows that
AB ≅ CD
Problem 2 :
In the diagram shown below,
A is the midpoint of MT
A is the midpoint of SR
Prove that MS  TR.
Plane for Proof :
Prove that ΔMAS ≅ ΔTAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that ∠M ≅ ∠T. Because these angles are formed by two segments intersected by a transversal, we can conclude that MS  TR.
Solution :
Statements A is the midpoint of MT A is the midpoint of SR MA ≅ TA, SA ≅ RA ∠MAS ≅ ∠TAR ΔMAS ≅ ΔTAR aaaaaaaa∠M ≅ ∠Taaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa MS  TR

Reasons Given Given Definition of midpoint Vertical Angles Theorem SAS Congruence Postulate Corresponding parts of congruent triangles are congruent. Alternate Interior Angles Converse 
Problem 3 :
In the diagram shown below,
∠1 ≅ ∠2
∠3 ≅ ∠4
Prove that ΔBCE ≅ ΔDCE.
Plane for Proof :
The only information we have about ΔBCE and ΔDCE is that ∠1 ≅ ∠2 and that CE ≅ CE. Notice, however, that sides BC and DC are also sides of ΔABC and ΔADC. If we can prove that ΔABC ≅ ΔADC, we can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ΔBCE and ΔDCE.
Solution :
Statements ∠1 ≅ ∠2 ∠3 ≅ ∠4 AC ≅ AC ΔABC ≅ ΔADC BC ≅ DC CE ≅ CE ΔBCE ≅ ΔDCE 
Reasons Given Given Reflexive Property of Congruence ASA Congruence Postulate Corresponding parts of ≅ Δ are ≅ Reflexive Property of Congruence SAS Congruence Postulate 
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