Chapter 8 Quadrilateral

Concept I Angle sum property of quadrilateral is 360^o

Q.1. Three angles of a quadrilateral are respectively edual to 110^o,40^o,and 50^o.find the fourth angle.

Q.2. In a quadrilateral ABCD,the angles A,B,C and D are in the ratio 1:2:3:4. Find the measure of each angle of the quadrilateral.

Q.3. In a quadrilateral ABCD, AO and BO are the bisectore of ∠A and ∠B, respectively. Prove that ∠AOB =1/2(∠C+∠D).

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Q.4. The sides BA and DC of a quadrilateral ABCD are produced as shown in fig 1. prove that a+b=x+y.                                                                                                                                                   

Q.5. In fig 2. The bisectors of  ∠B and ∠D of a quadrilateral ABCD meet CD and  AB produced at P and Q respectively. Prove that                   

∠P+∠Q=1/2(∠ABC+∠ADC).                                                                                                

Q6.In fig.3.  Dertermine ∠A+∠B+∠C+∠D+∠E.

Concept II Properties of a Parallelogram

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Q.1. In fig.4,ABCD is a parallelogram. Calculate the values of x and y.

Q.2.In a parallelogram ABCD,prove that sum of any two   4y consecutive angles is 180^o.                                          10x

Q.3.In a parallelogram, show that the bisectors of any two consecutive angle intersect at right angle.

Q.4.prove that the angle bisector of a parallelogram forms a rectangle.

Q.5. ABCD is a parallelogram. L and M are points on AB and DC respectively and AL=CM. Prove that LM and BD bisect each other.

Q.6. The diagonals of a parallelogram ABCD intersect at O. A line through AB at X and DC at Y. prove that OX=OY.

Q.8.ABCD is a parallelogram. AB is produced to E so that BE=AB. Prove that ED bisect BC.

Q.9.In a parallelogram ABCD, the bisector of ∠A also bisects BC at X . prove that AD=2AB.

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Q.11.In fig.6, PQRS is a parallelogram. PO and QO are respectively, the angle bisectors of ∠P and ∠Q. line LOM is drawn parallel to PQ. Prove that (i) PL = QM (ii) LO =OM.

Q.12 ABCD is a parallelogram and line segments AX and CY bisect the angles A and C respectively. Show that AX parallel to CY.

Q.13 the diagonals of a parallelogram ABCD intersect at O. A line through O intersects AB at X and DC at Y. prove that OX =OY.

Q.14.PQRS is a parallelogram. PX and QY are respectively the perpendiculars from P and Q to SR and RS produced. Prove that PX =QY.

Q.15. In ∆ABC lines are drawn through A,B and C parallel respectively to the sides BC,CA and AB, forming ∆PQR.show that BC =1/2QR.

Concept III Properties of a Converse Parallelogram

Q.1 In fig 7,ABCD is a parallelogram and X,Y are mid-points of sides AB and DC respectively. Show that quadrilateral AXCY is a parallelogram.

Q.2 In fig 8, ABC is an isosceles triangle in which AB=AC. CP//AB and AP is the bisector of exterior ∠CAD of ∆ABC. Prove that ∠PAC = ∠BCA and ABCP is a parallelogram.

Q.3 In a triangle ABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.

Q.4 In fig.9, BE┴AC.AD is any line from A to BC intersecting BE in H. P,Q and R are respectively the mid points of AH,AB and BC. Prove that ∠PQR= 90^O.

Concept IV  Midpoint theorem of  triangle

Q.1 If P,Q and R are respectively the mid- points of BC,CA and AB of an equilateral triangle ABC, prove that PQR is also an equilateral triangle.

Q.2. In a triangle ABC,E and F are mid points of AC and AB, respectively. the altitude AP to BC intersects FE at Q. prove that AQ=QP.

Q.3.let ∆ABC is an isosceles triangle with AB= AC and let P,Q and R the mid points of BC,CA and AB respectively. Show that AP⊥RQ and AP is bisected by QR.

Q.4. Prove that four triangle formed by joining mid-points of three sides in pairs, are congruent to each other.

Q.5. In a triangle ABC, p,q and r are the mid-points of sides BC,CA and AB respectively If AC= 21cm,BC= 30cm and AB= 29cm, find the perimeter of quadrilateral ARPQ.

Q.6 In fig 10, AD and BE are medians of ∆ABC and BE//DF. Prove that CF = ¼AC.

Q.7.P,Q and R are respectively, the mid-points of sides BC,CA and AB of ∆ABC. PR and BQ meet at X, CR and PQ meet at Y. Prove that XY =1/4BC.

Q.8.In fig 11,AD is the median through A in ∆ABC. E is the mid- point of AD. BE produced meet AC in F. prove that AF=1/3AC.

Q.9.ABCD is a parallelogram. P is a point on AD such that AP= 1/3AD and Q is a point on BC such that CQ = 1/3BC. Prove that AQCP is a parallelogram.

Q.10.P is the mid-point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R. prove that (i) AR= 2BC (ii) BR =2BQ.

Q.11. in fig 12, ABCD is a trapezium in which side AB is parallel to side DC and E is the mid-points of AD. If F is a point on the side BC such that BF=FC, prove that(i) EF//AB and (ii) EF =1/2(AB+CD)

Q.12 Prove that the line segment joining the mid- points of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half of difference of these sides.

Q.13.In fig12,ABCD is a trapezium in which AB//DC and AD =BC. If P,Q,R and S be respectively the mid- points of BA,BD,CD and CXA respectively .show that PQRS is A rhombus