Question

Given: AD ≅ BC and AD ∥ BC Prove: ABCD is a parallelogram. Statements Reasons 1. AD ≅ BC; AD ∥ BC 1. given 2. ∠CAD and ∠ACB are alternate interior ∠s 2. definition of alternate interior angles 3. ∠CAD ≅ ∠ACB 3. alternate interior angles are congruent 4. AC ≅ AC 4. reflexive property 5. △CAD ≅ △ACB 5. SAS congruency theorem 6. AB ≅ CD 6. ? 7. ABCD is a parallelogram 7. parallelogram side theorem What is the missing reason in step 6?

Answer 1

The missing reason in step 6 of the proof is 'Corresponding Parts of Congruent Triangles are Congruent (CPCTC)', which indicates AB ≅ CD as a result of the congruency between triangles CAD and ACB.

Step-by-step explanation:

The missing reason in step 6 of the proof that given AD ≅ BC and AD ∥ BC it can be proved that ABCD is a parallelogram, lies in the properties of congruent triangles.

When triangles CAD and ACB are proven congruent by the SAS congruency theorem (step 5), it implies that all corresponding sides and angles of the triangles are congruent.

Therefore, AB ≅ CD as these are corresponding parts of congruent triangles (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Once it is established that both pairs of opposite sides (AD & BC, AB & CD) are congruent, we can conclude that ABCD is indeed a parallelogram, based on the definition of a parallelogram which requires both pairs of opposite sides to be congruent (step 7).