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Theorems from Geometry:
# Section I
a)Prove that diagonals of a parallelogram divide it into two triangles of equal area.
b) Prove that diagonal PR of a given parallelogram PQRS divides two equal triangles
PQR and PSR.
c) Prove that, diagonal MO of given parallelogram MNOP divides two equal triangles
MNO and MPO.
# Section II
a)Prove that, parallelograms on the same base between the same parallels are equal
in area.
b) Parallelograms ABCD and EBCF standing on the same base BC between the same
parallels BC and AF are equal in area prove.
c) Parallelograms PQRS and MQRN standing on the same base QR between the same
parallels QR and PN are equal in area prove.
d) Prove that, rectagle MNPQ and parallelogram BNPA standing on the same base
NP between the same parallels NP and MA are equal in area.
e)Prove that, rectangle BNPA and parallelogram MNPQ are standing on the same
base NP between the same paralles NP and BQ equal in area..
f) In the given figure prove that
area of rectangle MNPF = area of
parallelogram ENPQ and area of
triangle MNE = area of triangle
FPQ.
g) In the given figure, prove that area of rectangle ABED = area of parallelogram
ABC area of triangle DEF.
CBEF and area of triangle
h) In the given figure prove
i) DAF = CBE.
ii) ABCD = ABEF.
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that
i) In the given figure prove that,
i) DXM = ZYN
ii) DXYZ = MXYN.
# Section III
a) Prove that the area of triangle is equals to half of the area of parallelogram
standing on the same base and between the same parallels.
b)Prove that the aera of triangle is equals to twice the area of triangle standing on
the same base and between the same parallels.
c) Prove that area of triangles MPQ is equal to one half of the area of parallelogram
RPQN standing on the same base PQ and between the same parallels PQ and MN.
d)Prove that area of paralegal RMNQ is equals to the twice the area of triangle PMN
standing on the same base MN and between the same paralles MN and PQ.
# Section IV
a) Prove that, trianlge standing on the same base between the same paralles are
equal in area.
b)Prove that area of PEF and QEF standing on the same base EF and between the
same parallels EF and PQ are equal in area.
c) Prove that area of EPQ and FPQ standing on the same base PQ and between the
same parallels PQ and EF are equal in area.
# Section V
a) Prove that, central angle is double of inscribed angle if they stand on the same are
in a circle.
b)Prove that inscribed angle is half of the central angle if they stand on the same are
in circle.
c) P, Q and R are the three point lies on the circumference of circle with centre A
prove that QAR is double of QPR ofter joining PQ, PR AQ and AR
d)L, M and N are the three points lies on the circumference of circle with centre B
prove that MBN is double of MLN after joining LM, LN BM and BN.
# Section VI
a) Prove that inscribed angle standing on the same are in a circle area equal.
b)Prove that angles on the same segment of a circle area equal.
c) PQMN is a circle with centre R after joining PN, PM, MQ and NQ prove that PNQ
is equal to PMQ.
d) WXYZ is a circle with centre A after WZ, WY, XY and XZ prove that WZX is equal
to WYX.
# Section VII
a) Prove that opposite angles of cyclic quadrilateral are supplementary.
b)Prove that Sum of opposite angles of cyclic quadrilateral are 1800 or two right
angles.
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c) PQRS is a cyclic quadrilateral with centre M prove that P + R = 1800 and Q +
S = 1800.
d)ABCD is a cyclic quadrilateral with centre N prove that A + C = 1800 and B +
D = 1800.
e) W, X, Y & Z are 4 points lies on the circumference of a circle with centre P. After
joining WX, XY, YZ & ZW, Prove that WXY and WZY are supplementary.
f) W, X, Y & Z are 4 points lies on the circumference of a circle with centre Q. After
joining WX, XY, YZ & ZW, Prove that XWZ and XYZ are supplementary.
# SectionVIII
a) Prove that angle on the semicircle being right angle.
# Section IX
a) Prove that if one side of a cyclic quadrilateral is produced, prove that the exterior
angle so formed is equal to the opposite interior angle of quadrilateral.
# Section X
a) Prove that the length of two tangents to a circle at the point of contact from a
external point are equal.
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