Download Quadrilaterals

1O
Geometry I - Rectilineor Figures
C. Triangle inequality
The sum of the lengths of any two sides of a triangle is greater than the length
the
e.g.
third side. This relation
is known as
of
triangle inequalitl'.
In AABC, a+b> c,and
b+c>a,and
c+a>b.
\&'e can apply the above result
to determine rvhether three line segments with given lengths can form
a
triangle.
rc.7 Quadrilaterals
A. Properties of parallelograms
A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
If ABCD is a parallelogram, then
AB=DC and AD=BC
lA= lC
(Opp.Sides of//gram)
(opp. Zs of
Ar-------l'--
and
ll
lB = lD
(iii)AO=OC and BO=OD
gram)
ll
gram)
D
ハ
ハ
D
(diags. of
ー
0
ー
B. Conditions for parallelogramr
(i)
If PQ = SR and P$ = QR, then
PQRS is a parallelogram.
(opp. sides equal)
(iii) If PO = OR and QO =
(ii) If ZP= lRand lQ= ZS,then
PQRS is a parallelogram.
(opp. Zs equal)
PQRS is a parallelogram.
alld PS〃 QR,then
PQRS is a paraudogram.
(diags. bisect each other)
(Opp.Sides equal and//)
OS,
then
(市 )IfPS=QR
0
Note:
The proofs related to parallelograms belong to the non-foundation part of the syllabus.
ang e nequal性
y三 角不等式
paral e ogram平 イ
子四邊形
163
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Geometry I - Rectilineor Figures
c.
Properties of rhombuses, rectangtes, squares, trapeziums and kites
(a)
Rhombuses
A
(i)
is a parallelogram with four equal sides
It has all properties of a parallelogram.
(ii) All
its sides are equal in length.
(iii) Its diagonals
are perpendicular to each other.
(iv) Its interior angles are bisected by the diagonals.
(Abbreviation: property of rhombus)
(b)
Rectangles
A rectangle is a parallelogram with four interior right angles.
(i) It has all properties of a parallelogram.
(i0 All the interior
angles are right angles.
(iii) Its diagonals are equal in length.
(rv) Its diagonals bisect each other into four equal parts.
(Abbreviation: property of rectangle)
(C)
Squares
A
(i)
It has all properties of a rectangle.
(ii)
It has all properties of a rhombus.
is a rectangle with four equal sides.
{
A square is both a
rhombus and a
rectangle.
(iii) Each angle between a diagonal and a side is 45o.
(Abbreviation: property of square)
(d)
Trapeziums
A trapezium is a quadrilateral with only one pair of paraliel opposite
trapezium are:
(i)
sides. Two special kinds
A trapezium with one of its sides perpendicular to the t.,yo bases is called
a right-angled trapezium.
.
(ii) A trapezium with non-parallel
sides of equal length
following properties:
∠P=∠ S and∠ Q=∠ R
P
rhombus麦 形
"
1要
PR=QS
s
rectang e長 方形
isosceles trapezium等
′‐
called an isosceles trapezium. It has the
梯形
square E)1fu
lght― angttd
trapezium直 角梯形
tO
Geometry I - Rectilineor
(e)Kites
A
is a quadrilateral with two pairs ofequal attacent sides.
Consider kite ABCD,where AB=AD and CB=CD.AC intersects BD at O.
￨￨
It has the follo、 ving properties:
(i)∠ ABC=∠ ADC
(ii)BOth∠ BAD and∠ BCD are bisected bv AC.
(ili)BO=D0
(市 )AC
tt BD
Ｄ Ｃ
10.8 Mid-point Theorern and lntercept Theorenn
A. Mid-point theorem
In AABC,
1f
AM = MB andAN = NC, then
(i) MN il BC,
(i0
I
MN =-BC.
2
(mid-pt. theorem)
B. Intercept theorem
then DE=EF.
In△ ABC,ifAE=EB and
EF〃 BC,then AF=FC.
(intercept theorelln)
(intercept theorem)
If AD//BE//CF and AB=BC,
S Of
0
s the
Example饉 : In the igure,A3//CD,∠ 34K=30° ,∠ AKD=130° 狙 d
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