Properties of a Parallelogram and Related Theorems - Quadrilaterals, Class 9, Mathematics | Extra Documents & Tests for Class 9 PDF Download

ANGLE SUM PROPERTY OF A QUADRILATERAL

THEOREM-I : The sum of the four angles of a quadrilateral is 360°.
Given : A quadrilateral ABCD.
To Prove : ∠A + ∠B + ∠C + ∠D = 360°

Construction : Join AC. 
Proof :

Hence, proved.

Ex.1 Three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.
Sol. Let the measure of the fourth angle be x°.
∴  56° + 100° + 88° + x° = 360°  ∴ [Sum of all the angles of quadrileteral is 360°]
⇒  244+x=360 
⇒   x = 360 – 244 = 116
Hence, the measure of the fourth angle is 116°.

Ex.2 The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
Sol. Let the four angles of the quadrilateral be 3x, 5x, 9x and 13x . [NCERT]
∴  3x + 5x + 9x + 13x = 360°      ∴ [Sum of all the angles of quadrileteral is 360°]
⇒ 30x = 360°
⇒ x =12°
Hence, the angles of the quadrilateral are 3 × 12° = 36°, 5 × 12° = 60°, 9 × 12° = 108

PARALLELOGRAM:- A quadrilateral in which both pairs of opposite Csides are parallel is called a parallelogram.

In the figure, ABCD is a quadrilateral in which AB ⊥ DC, BC ⊥ AD.
∴ quadrilateral ABCD is a parallelogram.

PROPERTIES OF A PARALLELOGRAM

THEOREM-1 A diagonal of a parallelogram divides it into two congruent triangles.
Given : ABCD is a parallelogram and AC is a diagonal which forms two triangles CAB and ACD.
To Prove : ΔACD ≌ ΔCAB

Proof :

Hence, proved.

THEOREM-2 In a parallelogram, opposite sides are equal.

Given : ABCD is a parallelogram.
To Prove : AB = CD and BC = DA

Construction : Join AC.
Proof :

Hence, proved.

THEOREM-3 In a parallelogram, opposite angles are equal.

Given : ABCD is a parallelogram.
To Prove : ∠A = ∠C and ∠B = ∠D
Proof :

Hence proved.

THEOREM-4 : The diagonals of a parallelogram bisect each other.

Given : ABCD is a parallelogram, diagonals AC and BD intersect at O. O
To Prove : OA = OC and OB = OD
Proof :

Hence, proved.