Question

2) Prove that opposite sides and opposite angles of a parallelogram are congruent.​

Answer 1

Answer:

If a quadrilateral has 2 sets of opposite sides congruent, then it is a parallelogram. THEOREM: If a quadrilateral has 2 sets of opposite angles congruent, then it is a parallelogram. THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram.

Step-by-step explanation:

Properties of parallelograms

1.Opposite sides are congruent (AB = DC).

2.Opposite angels are congruent (D = B).

3.Consecutive angles are supplementary (A + D = 180°).

4.If one angle is right, then all angles are right.

5.The diagonals of a parallelogram bisect each other.

6.Each diagonal of a parallelogram separates it into two congruent triangles.

Answer 2

Given : opposite sides and opposite angles of a parallelogram are congruent

To Find : Prove

Solution:

Parallelogram  , Quadrilateral in which opposite sides are parallel

Take ABCD as parallelogram

AB || CD  and BC || AD

Join AC

Check Triangle

ΔABC  and ΔCDA

∠BAC  = ∠DCA  ( ∵ AB || DC and AC is transversal  , alternate interior angles)

∠BCA  = ∠DAC  ( ∵ AD || BC and AC is transversal  , alternate interior angles)

AC ≅ AC   reflexive property

Hence ΔABC ≅ ΔCDA   using ASA congruence

Corresponding sides and angles of congruent triangles are congruent

AB ≅ CD   and BC ≅ AD  so opposite sides are congruent

∠ABC ≅  ∠CDA  => ∠B ≅   ∠D  opposite angles are  congruent

Similarly can be shown ∠A ≅ ∠C  by joining BD

QED

Hence proved

opposite sides and opposite angles of a parallelogram are congruent

Additional Info:

Diagonals of parallelogram bisect each other

Adjacent angles are supplementary. ( adds up to 180°)

Properties of angles formed by transversal line  with two parallel lines :

• Corresponding angles are congruent. ( Equal in Measure)

• Alternate angles are congruent.  ( Interiors & Exterior  both )  

• Co-Interior angles are supplementary. ( adds up to 180°)

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