4.In the diagram, ABCDis a trapezium with AB = CDProve that (i) BADis congruent to CDA(ii)
.
.
,
,
s)
)
BC
)
)
.

24
5.
In the diagram,
BCDE
is a rectangle.
AED
is isosceles with
AD = AE
.
AFB
and
AGC
are straight lines.
Show that
(i)
ABE
is congruent to
ACD
,
(ii)
AEF
is congruent to
ADG
.
(Pg256Q13)
Proof
(i)
AED
=
ADE
(base
s of isos.
AED
)
BEF
=
CDG
= 90
(
s of rectangle)
AED
+
BEF
=
ADE
+
CDG
Hence,
AEB
=
ADC
.
AE
=
AD
(given)
BE
=
CD
(opp sides of rectangle)
Hence
ABE
is congruent to
ACD
. (SAS)
(ii)
EAF
=
DAG
(corr
s of congruent
s)
AE
=
AD
(given)
AEF
=
ADG
(base
s of isos
)
Hence
AEF
is congruent to
ADG
. (ASA)
6.
In the diagram,
ABCD
and,
CDEF
are parallelograms. The point
X
is the mid-point of
AC
and
of
BD
, and the point
Y
is the mid-point of
CE
and of
DF
.
Prove that
(i)
AE
is parallel to
XY
,
(ii)
AE
is parallel to
BF
,
(iii)
BAEF
is a parallelogram.
(Pg256Q14)
Proof
(i)
AE
//
XY
(Mid-point Theorem)
(ii)
XY
//
BF
(Mid-point Theorem)
Hence,
AE
//
XY
//
BF
.
(iii)
AB
//
CD
and
CD
//
EF
Hence,
AB
//
EF
.
BAEF
is a parallelogram.
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25
7.
In the diagram below,
ABCD
is a quadrilateral.
AE = BE
and
BF = FC
. The diagonal
AC
intersects
DE
at
X
and
DF
at
Y
, so that
AX = XY = YC
.
Prove that
(i)
BXDY
is a parallelogram,
(ii)
AXB
is similar to
CYD
.
(iii)
ABCD
is a parallelogram.
(Pg270Q5)
Proof
(i)
EX
//
BY
(Mid-point Theorem)
FY
//
BX
(Mid-point Theorem)
Hence,
BXDY
is a parallelogram.
(ii)
AXB
=
XYF
(corr.
s,
BX
//
FY
)
=
CYD
(vert. opp.
s)
AX
=
CY
(given)
BX
=
DY
(opp. sides of parallelogram)
AXB
is congruent to
CYD
.
(SAS)
(iii)
BAX
=
DCY
.
(corr
s of congruent
s)
AB
//
CD
(alt
s are equal)
AB
=
CD
(corr sides of congruent
s)
ABCD
is a parallelogram.
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26
Singapore Chinese Girls’
School
Secondary 4
Additional Mathematics
Plane Geometry
Name:
(
)
Class: Sec 4_____
Date: ________________
Worksheet 7: Angle Properties of Circles
Some Angle Properties of Circles
Property
Abbreviation
1.
An angle in a semicircle is a right angle.
in a semicircle
2.
An angle at the centre is twice any angle at the
circumference.
at centre = 2
at
ce
3.
Angles in the same segment are equal.
s in the same segment
4.
The exterior angle of a cyclic quadrilateral is equal to the
interior opposite angle.
Ext
of a cyclic quad
5.
Angles in opposite segments are supplementary.
x
+
y
= 180
s in opp segments
(or opp
s of a cyclic quad)
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