# Class 9 Quadrilaterals Important Questions

Important Questions**Quadrilaterals Important Questions**

**Show that each angle of a rectangle is a right angle.****Find all the angles of a parallelogram if one angle is 60°.****Two parallel lines l and m are intersected by a transversal p (See in fig.). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.****The angles of quadrilateral are in the ratio 2 : 4 : 8 : 12. Find all the angles of the quadrilateral.****The diagonals of which quadrilateral are equal and bisect each other at 90°.****If the diagonals of a parallelogram are equal, then show that it is a rectangle.****In a trapezium ABCD, AB∥CD. Calculate ∠C and ∠D if ∠A = 55° and ∠B = 70°**.**L, M, N, K are mid-points of sides BC, CD, DA and AB respectively of square ABCD, prove that DL, DK, BM and BN enclose a rhombus.****In a parallelogram ABCD find the measure of all the angles if one its angles is 15° less than twice the smallest angle.****If ABCD is a parallelogram, then prove that ar(∆ABD) = ar(∆BCD) = ar(∆ABC) = ar(∆ACD) = ½ar(||**^{gm}ABCD)**If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in area. If G is the mid point of median AD, prove that ar(∆BGC) = 2ar(∆AGC).****ABCD is a trapezium in which AB || CD and AD = BC (see Fig.). Show that****∠ A = ∠ B****∠ C = ∠ D****∆ ABC ≅ ∆ BAD****diagonal AC = diagonal BD**

[Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

**Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that: ar(∆APB) x ar(∆CPD) = ar(∆APD) x ar(∆BPC).****In a ∆ABC, P and Q are respectively, the mid-points of AB and BC and R is the mid-point of AP. Prove tha**t**ar(∆PBQ) = ar(∆ARC)****ar(∆PRQ) = ½ ar(∆ARC)****ar(∆RQC) = ⅜ ar(∆ABC).**

**D is the mid-point of side BC of ∆ABC and E is the mid-point of BD. If O is the mid-point of AE, prove that ar(∆BOE) = ⅛ ar(∆ABC).****In ∆ ABC, D, E and F are respectively the mid-points of sides AB, BC and CA (see Fig.). Show that ∆ ABC is divided into four congruent triangles by joining D, E and F.****ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that****D is the mid-point of AC****MD ⊥ AC****CM = MA =½AB**.

**In the figure, ABC is a right triangle right angled at A, BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment AX ⊥ DE meets BC at Y. Show that****∆MBC ≅ ∆ABD****ar(BYXD) = ar(∆MBC)****ar(BYXD) = ar(ABMN)****∆FCB ≅ ∆ACE****ar(CYXE) = 2ar(∆FCB)****ar(CYXE) = ar(∆CFG)****ar(BCED) = ar(AMBN) + ar(ACFG)**

**ABCD is a quadrilaeral in which the bisectors of ∠A and****∠**C meet DC produced at Y and BA produced at X respectively. Prove that**∠**X +**∠**Y = ½(**∠**A +C).**∠****ABCD is a parallelogram. P is the mid-point of AB. BD and CP intersect at Q such that CQ : QP = 3 : 1. If ar(∆PBQ) = 10 cm**^{2}, find the area of parallelogram ABCD.**A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle and parallelogram is****1 : 1****1 : 2****2 : 1****1 : 3**

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