# 9th Class Mathematics Quadrilaterals Important Points

## Important Points

Category : 9th Class

### Important Points

(i) A quadrilateral is a parallelogram if their opposite sides are equal.

(ii) A quadrilateral is a parallelogram if their opposite angles are equal.

(iii) If the diagonal of a quadrilateral bisect each other then it is a parallelogram.

(iv) A quadrilateral is a parallelogram if its one pair of opposite sides are equal and parallel.

(v) The diagonals of rectangle are equal.

(vi) If the two diagonals of a parallelogram is equal then the parallelogram is a rectangle.

(vii) The diagonal of the rhombus are perpendicular bisector of each other.

(viii) A parallelogram is a square if the diagonals of a parallelogram are equal and bisector at right angle.

Intercept Theorem

If a transversal intersect three and more than three parallel lines in such a way that all the intercepts are equal then the intercept on any other transversal is also equal.

Given:

Three parallel lines a, b, c intersected by a transversal $x$ at L, M, N respectively so that LM = MN. Another transversal y cutting a, b, c at T, U, V respectively

Proof:

Since $\text{LT }\!\!|\!\!\text{ }\,\text{ }\!\!|\!\!\text{ MU}$ and $~\text{LM }\!\!|\!\!\text{ }\,\text{ }\!\!|\!\!\text{ TU}$

$\therefore$ LMUT is a parallelogram $\therefore$ TU = LM                                                     .....(i)

Similarly

MNVU is a parallelogram

$\therefore$ MN=UV                                                    .....(ii)

But $\therefore$ TU = UV

Now In

$\Delta \text{MLT}$and $\Delta \text{TUM}$          (Alternate)

$\angle \text{MTL}=\angle \text{MTU}$                                (Opposite angle of Parallelogram)

LT = UM                                        (Opposite sides of Parallelogram)

MT=MT                                (Common)

$\therefore$ By SOS

$\Delta \text{MTL}\cong \Delta \text{TUM}$

By CPCT,

LM=TU

Similarly

MN=UV But

LM=MN

$\therefore$ TU = UV

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