Parallelograms

Quadrilaterals A polygon with four sides is called a quadrilateral. B C D A polygon with four sides is called a quadrilateral. In a quadrilateral, (a) any two sides that have a common vertex are called adjacent sides, (b) any two non-adjacent sides are called opposite sides, e.g. AB and BC are a pair of adjacent sides with common vertex B, e.g. AD and BC are a pair of opposite side without a common vertex,

Quadrilaterals A polygon with four sides is called a quadrilateral. B C D A polygon with four sides is called a quadrilateral. In a quadrilateral, (c) any two interior angles that are not lying on the same side are called opposite angles, (d) the sum of all the interior angles is 360. e.g. A and C are a pair of opposite angles, sum of interior angles of a quadrilateral = (4 – 2)  180

Properties of Parallelograms A parallelogram is a quadrilateral that has two pairs of parallel opposite sides. What are the properties of a parallelogram?

The opposite sides of a parallelogram are equal. B C D Property I AD = BC The opposite sides of a parallelogram are equal. i.e. AB = DC, [Abbreviation: opp. sides of // gram] A B C D Property II i.e. A = C, B = D The opposite angles of a parallelogram are equal. [Abbreviation: opp. s of // gram]

[Abbreviation: diags. of // gram] C D Property III O The diagonals of a parallelogram bisect each other. i.e. AO = OC, BO = OD [Abbreviation: diags. of // gram]

Can you find the unknowns in the figure? B C D 30 y 7 cm x cm In the figure, ABCD is a parallelogram. = AB DC ∴ (opp. side of // gram) = 7 x  = Ð + 180 C B ∵ (int. s, AB // DC)  = + 180 30 y ∴  = 150 y

Follow-up question Find the unknowns in the figure. Solution 4 cm Find the unknowns in the figure. A B C D 2x – 15 b cm (7 – b) cm 75 Solution a cm ∵ BC = AD (opp. sides of // gram) 4 = a ∴ ∵ AB = DC (opp. sides of // gram) 7 - = b ∴ 7 2 = b 5 . 3 = b

Follow-up question (cont’d) 4 cm Find the unknowns in the figure. A B C D 2x – 15 b cm (7 – b) cm 75 Solution a cm ∵ B = D (opp. s of // gram) ∴  - = 15 2 75 x  = 90 2 x  = 45 x

Identifying Parallelograms How can you determine whether a quadrilateral is a parallelogram? Very good! Can you think of any other conditions to identify a parallelogram? By the definition of a parallelogram, if the quadrilateral has two pairs of parallel opposite sides, it is a parallelogram.

Both pairs of opposite sides are equal. and AD = BC, i.e. if AB = DC Condition I A B C D Both pairs of opposite sides are equal. and AD = BC, i.e. if AB = DC then ABCD is a parallelogram. [Abbreviation: opp. sides equal ] Condition II A B C D Both pairs of opposite angles are equal. i.e. if A = C and B = D, then ABCD is a parallelogram. [Abbreviation: opp. s equal ]

The diagonals bisect each other. i.e. if AO = OC and BO = OD, Condition III The diagonals bisect each other. O i.e. if AO = OC and BO = OD, then ABCD is a parallelogram. [Abbreviation: diags. bisect each other ] A B C D Condition IV One pair of opposite sides are equal and parallel. i.e. if AD = BC and AD // BC, then ABCD is a parallelogram. [Abbreviation: opp. sides equal and // ]

In the figure, ABCD is a quadrilateral. AB = DC = 3 cm ∵ B + C = 75 + 105 = 180 ∴ AB // DC ∵ AB = DC and AB // DC ∴ ABCD is a parallelogram. A B C D 75 105 3 cm given int. s supp. opp. sides equal and //

Follow-up question The figure shows a quadrilateral ABCD. (a) Find a. (b) Prove that ABCD is a parallelogram. 55 2a + 15 70 + a a B C Solution (a) ∵ Sum of the interior angles of a quadrilateral is 360. ∴ (2a + 15) + a + (70 + a) + 55 = 360 4a + 140 = 360 4a = 220 a = 55

Follow-up question (cont’d) A D The figure shows a quadrilateral ABCD. (a) Find a. (b) Prove that ABCD is a parallelogram. 55 2a + 15 70 + a a B C Solution A = 2a + 15 = 2(55) + 15 = 125 B = a = 55 C = 70 + a = 70 + 55 = 125 D = 55 ∴ A = C and B = D ∴ ABCD is a parallelogram. given opp. s equal