Problem 1 :
In the diagram shown below, AB  CD, BC  DA. Prove 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.
Problem 3 :
In the diagram shown below,
∠1 ≅ ∠2
∠3 ≅ ∠4
Prove that ΔBCE ≅ ΔDCE.
1. Answer :
Plane for Proof :
Show that ΔABD ≅ ΔCDB. Then use the fact that corresponding parts of congruent triangles are congruent.
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
2. Answer :
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.
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 
3. Answer :
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.
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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