Question

Given: AB is congruent to CD and BC is congruent to AD Prove: AB ll CD and BC ll AD

Answer 1

Given

• A quadrilateral ABCD with opposite pairs of sides being congruent.

To prove

• Opposite sides are parallel.

Solution

We'll prove first the triangles formed by diagonal are congruent then we'll prove the angles on either side of diagonal are congruent.

The steps are:

• Step  ⇒   Statement  ⇒   Reason

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• 1   ⇒  AB ≅ CD and BC ≅ AD     ⇒     Given
• 2  ⇒  AC = CA                              ⇒    Common side
• 3  ⇒  ΔABC ≅ ΔCDA                   ⇒    Side-side-side congruence
• 4  ⇒  ∠BAC ≅ ∠DCA                   ⇒    Corresponding angles of congruent triangles
• 5  ⇒  ∠BAC and ∠DCA are alternate interior angles  ⇒  Definition of alternate interior angles
• 6  ⇒  AB║CD                               ⇒   The converse of alternate interior angles theorem
• 7  ⇒  ∠BCA ≅ ∠DAC                   ⇒   Corresponding angles of congruent triangles
• 8  ⇒  ∠BCA & ∠DAC are alternate interior angles  ⇒  Definition of alternate interior angles
• 9  ⇒ BC║AD                                ⇒   The converse of alternate interior angles theorem