RD Sharma Solutions for Class 9 Maths Chapter 14 – Quadrilaterals

Important Theories

• The sum of all four quadrilateral angles is 360 °.

• The diagonal of the parallelogram divides it into two interlocking triangles.

• A quadrilateral is a parallelogram if

o   The opposite sides are equal.

o   The opposing angles are equal,

o   Diagonals are divided into two.

• A quadrilateral is a parallelogram if

o   Its opposing angles are equal.

o   Its opposite sides are equal.

o   Its diagonals are split in two.

• A pair of opposite sides are equal and parallel.

• Rectangular diagonals are divided into two and are equal and opposite.

• The rhombus diagonals are separated by right angles and are not evenly spaced.

• Square diagonals divide the other two into right angles and are equal and vice versa.

• A part of a line connecting points between any two sides of a triangle corresponds to a third and is equal to half of it. (Mid-point theorem)

• A line drawn in the centre of one side of the triangle, in line with the other side, crosses the third side in its centre. (Speaking of mid-point theorem)

• A quadrilateral formed by combining points between quadrilateral sides, taken in sequence, parallelogram.

• In a quadrilateral, if the diagonals divide by two, then form a parallelogram.

Properties of a Parallelogram:

• A diagonal of a parallelogram, dividing it into parallel triangles.

• In parallelograms, the opposing sides are equal.

• In parallelograms, opposing angles are equal.

• The diagonal of the parallelogram is divided into two parts.

Angle Sum Property Quadrilateral

The sum of the four quadrilateral angles is 360 °. Angle Sum Property when we draw a quadrilateral diagonal, it divides it into two triangles.

We also know that the angle sum property of a triangle i.e. the sum of all the three angles of the triangle is 180 °.

Total angle ∆ADC = 180 °.

Total angle ∆ABC = 180 °.

By adding both we get ∠A + ∠B + ∠C + ∠D = 360 °

Thus, the sum of four quadrilateral angles is 360 °.

The Mid-point Theorem

• If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

• If AB = BC and CD = DE then BD ∥ AE.

• If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.

• If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

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