Proving the converse of the parallelogram side theorem

answer:

Determining if a Quadrilateral is a Parallelogram

Opposite Sides Theorem Converse: If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram. Opposite Angles Theorem Converse: If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.

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Let parallelogram ABCD, have sides such that AB is parallel to CD and AC is parallel to BD.

AB ll CD and AC ll BD --> Write the problem.

Draw line BD --> There is exactly one line through any two points.

angle ABD≅angle BDC --> Alternate Interior Angles Theorem.

angle ADB≅angle DBC --> Alternate Interior Angles Theorem.

BD=BD --> Reflexive Property of equality.

∆ABD≅∆BDC --> ASA Congruence Theorem.

AB=CD, AD=BC Corresponding Parts of Congruent Triangles are Congruent.

The answer is letter D. The congruent triangles are congruent.

A parallelogram is congruent if the corresponding angles are congruent and they form a common triangle. A parallelogram is also congruent if the sides are parallel to each other and can be divided into two common triangles.

To prove: In the ||gm ABCD, segment AB ≅ CD & segment BC ≅ AD.

Proof:

∵ ABCD is a parallelogram,

→ segment AB || segment DC 

→ segment BC || segment AD 

Now, the line AC is a transversal.

In  Δ ABC and ΔADC∠1 = ∠4 (alternate interior angles)

similarly, ∠2 = ∠3 (alternate interior angles)

AC is common.

Hence, by ASA congruency property, we can say that, 

Δ ABC ≅ ΔADC

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⇒ segment AB ≅ CD & segment BC ≅ AD (congruent parts of congruent triangles)

Proved.