Parallelogram

Parallelogram

Parallelogram

Definition: A quadrilateral having its opposite sides parallel is called a parallelogram. In the adjoining figure, ABCD and ADBC. So ABCD is a parallelogram.
Parallelogram ABCD

Square, Rectangle, Rhombus are some special types of parallelograms
Properties of parallelogram
1.       Opposite sides of a parallelogram are equal
2.       Opposite angles of a parallelogram are equal.
3.       Diagonals of a parallelogram bisect each other.

Proofs:

1.       Prove theoretically that the opposite sides of a parallelogram are equal.
ABCD is a parallelogram where, AB∥DC and BC∥AD
               Given: ABCD is a parallelogram where, ABDC and BCAD.
               To prove: AB=CD and AD=BC
               Construction: B and D joined.
               Proof:
                             Statements                          Reasons
1. In ΔABD and ΔBCD
 i.    ABD = BDC (A) ---------> Alternate angles
 ii.   BD = BD (S) -----------------> Common side
 iii. ADB = CBD (A) ---------> Alternate angles
2. ΔABD ΔBCD -------------------> By A.S.A. axiom
3. AD = CD and AD = BC ---------> Corresponding sides of congruent triangles
                                                                                                                     Proved.

2.       Prove theoretically that the opposite angles of a parallelogram are equal.
ABCD is a parallelogram where, AB∥DC and BC∥AD
               Given: ABCD is a parallelogram where, ABDC and BCAD.
               To prove: BAD = BCD and ABC = ADC
               Construction: B and D joined.
               Proof:
                            Statements                             Reasons
1.       In ΔABD and ΔBCD
    i.    ABD = BDC (A) -------> Alternate angles
    ii.   BD = BD (S) ---------------> Common side
    iii. ADB = CBD (A) -------> Alternate angles
2.       ΔABD ΔBCD -----------------> By A.S.A. axiom
3.       BAD = BCD ------------------> Corresponding angles of congruent triangles
4.       ABC = ADC ------------------> Similarly by joining A and C
                                                                                                      Proved.

3.       Prove theoretically that the diagonals of a parallelogram bisect each other.
ABCD is a parallelogram. Diagonals AC and BD intersect at O
               Given: ABCD is a parallelogram. Diagonals AC and BD intersect at O.
               To prove: AO=CO and BO=DO
               Proof:
                          Statements                                    Reasons
1.       In ΔAOD and ΔBOC
    i.   OAD = OCB (A) ------------> Alternate angles
    ii.   AD = BC (S) --------------------> Opposite sides of a parallelogram
    iii. ODA = OBC (A) ------------> Alternate angles
2.       ΔABD ΔBCD ----------------------> By A.S.A. axiom
3.       AO = CO and BO = DO -----------> Corresponding sides of congruent triangles
                                                                                                                          Proved.

Workout Examples

Example 1: Find the values of unknown angles in the given parallelogram.
Example 1 Parallelogram ABCD
Solution: From the figure,
                                              a + 120° = 180° ----------------> Co-interior angles.
                                or,          a = 180° - 120°
                                or,          a = 60°
                                              b = 120° --------------------------> Opposite angles of a parallelogram.
                                              c = a -----------------------------> Opposite angles of a parallelogram.
                                                 = 60°

                                            a = 60°
                                              b = 120°
                                              c = 60°


Example 2: Find the values of unknown angles in the given parallelogram.
Example 2 Parallelogram ABCD
Solution: From the figure,
                                              4a + 5a = 180° ----------------> Co-interior angles.
                                or,         9a = 180°
                                or,           a = 180°/9
                                or,           a = 20°
                                               b = 4a --------------------> Corresponding angles.
                                                   = 4 × 20°
                                                   = 80°
                                                c = b --------------------> Alternate angles.
                                                   = 80°
                                                d = 5a --------------------> Opposite angles of a parallelogram.
                                                   = 5 × 20°
                                                   = 100°

                                             a = 20°
                                               b = 80°
                                               c = 80°
                                               d = 100°


Example 3: Find the values of unknown angles in the given parallelogram.
Example 3 Parallelogram ABCD
Solution: From the figure,
                                              a + 35° + 60° = 180° ----------------> Sum of angles of ΔABC.
                                or,          a + 95° = 180°
                                or,          a = 180° - 95°
                                or,          a = 85°
                                              b = 35° --------------------------> Alternate angles.
                                              c = a -----------------------------> Alternate angles.
                                                 = 85°
                                              d = 60° --------------------------> Opposite angles of a parallelogram.
                                            a = 85°
                                              b = 35°
                                              c = 85°
                                              d = 60°


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