Math (11)_text.pdf

(1979 KB) Pobierz
PARALLELOGRAMS
AND
TRIANGLES
Theorem
In
a
parallelogram
(i)
(ii)
(iii)
Opposite
sides
are
congruent.
Opposite
angles
are
congruent.
The
diagonals
In
bisect
each
other.
Oiven
a
quadrilatéral
ABCD,
ABIIDC.BCIIAD
and
meet
each
other
Toi
Provt*
(i)
the
diagonals
AC,
BD
at
point
O.
(ü)
(iii)
AB=
DC.
AD
s
BC
ZADC
s
ZABC,
ZBAD
=
ZBCD
ÔÂs
OC.
OB=
ÔD
In
the
figure
as
shown,
Construction'
we
label
the
angles
as
Zl, Z2,
Z3,
Z4,
Z5
and
Z6.
Proof
Statements
(i)
Reasons
Alternate
angles
In
AABD
s
ACDB
Common
Alternate
angles
Z4
-
Zl
BD
BD
Z2
=
Z3
AABD
=
ACDB
So,
A.S.A.
=
A.S.A.
ÂB=
Since
DC.
AD
s
BC
(corresponding
sides
of
congruent
triangles)
(corresponding
angles
of
congruent
triangles)
and
(ü)
ZA
=
ZC
Zl
s
Z4
Z2
~
Z3
....(a)
Proved
Proved
and
m
....(b)
\
mZl
+
mZ2
=
mZ4
+
mZ3
From
(a)
and
(b)
or
or
mZADC
=
mZABC
ZADCs
ZABC
and
(iii)
ZBAD
=
ZBCD
In
AB
OC
O
ADOA
BCsÂD
Z5
=
Z6
Z3
=
Z2
ABOC
s
ADOA
Proved
in
(i)
Proved
in
(i)
Vertical
angles
Proved
A.A.S
=
A.A.S
Corresponding
triangles)
Hence
ÔC=ÔÂ,ÔB=ÔD
sides
of
congruent
Each
diagonal
of
a
parallelogram
bisects
it
into
two
congruent
triangles.
The
same
bisectors
of
two
angles
on
the
side
of
a
parallelogram
eut
each
other
at
right
angles.
ïïm'iT
A
parallelogram
ABCD,
in
which
AB
DC,
ÂDII
BC
II
The
bisectors
of
ZA
and
ZB
eut
each
other
at
E.
mZE
=
90°
<
'(instruction
Name
the
IJWffB
angles
ZI
and
Z2
as
shown
in
the
figure.
Statements
Reasons
mZ
1
+
mZ2
*
mZl=-mZBAD,
2
1
=
^
(mZB
AD
+
mZABC)
II
OC
mZ2=—
mABC
2
Int.angles
on
the
same
sideof
<
O
o
AAÜ
-90
Which
cuts
j |
AB
segments
AD
and
BC
aresupplementary.
Hence
in
AABE,
mZE
=
90°
mZl
+
mZ2
=90°
(proved)
EXERCISE
11.1
(1)
One
130°.
angle
of
a
parallelogram
is
its
Find
the
measures
of
remaining
angles.
ABCD
is
a
parallelogram
that
ni
Z
A
=
130°
(Required)
To
find
the
measures
of
ZB, ZC,
ZD
Reasons
Proof
Statcments
mZC
=
mZA
130°
mZC
=
mZB
+
mZA
=
Opposite
angles
of
parallelogram.
Given,
180°
f !
mZA
=
130°
is
AD
BC
Given
Ju
and
AB
tran
sversal
sum
of
interior
angles.
180°
mZB
+
130°
=
180°
-130°
mZB
=
50°
mZB
=
mZD
=
mZB
50°
mZD
=
mZB
=
50°,
mZC
=
130°,
mZD
=
50°
1
mZA
=
130°
Opp.
angles
As
mZB
=
50°
î
>
L.
One
exterior
angle
formed
on
producing
is
40°.
one
side
of
a
parallelogram
Find
the
measures
of
its
interior
angles.
ABCD
is
a
parallelogram,
side
AB
has
been
produced
to
p
to
form
exterior
angle
mZCBP
=
40°
Esn
To
and
name
the
interior
angles
as
ZI,
ZC,
ZD,
ZA.
find
the
degree
measures
of
Zl,
ZC,
ZD,
ZA
Reasons
Statements
mZl
+
mZCBP
mZi
+40“
180°
Supp.angies.
180°
mZCBP
=
40°
given
m
Z
mZl
rs
180°-
140°
O
o
(0
mZD
mZD
mZA
+
mZl
=
mZl
140°..
-..(ii)
Opp.angles
of
llm
From
(i)
=
=
180°
40°..
-
-
00
O
O
AD
1
1
BC
and
AB
is
transversal.
(Interior
angles)
mZA+
140°
mZA
=
mZA
=
mZC
mZC
Thus
180°
140°
From
(i)
—
(iii)
=
mZA
40°
Opp.
angles
=
=
From
(iii)
O
mZl
O
mZC
=
40°
—
uL.
Theorem
If
Given
two
opposite
sides
of
a
it
In
a
quadrilatéral
ABCD,
quadrilatéral
are
congruent
is
and
parallel,
ÀB
s
DCand
ABU
DC
a
parallelogram.
ABCD
Construction
is
a
parallelogram.
Join
the
point
figure,
B
to
D
and
in
the
name
the
angles
as
indicated:
ZI, Z2,
Z3
and
Z
4
Proof
Statements
In
Reasons
AABD
o
ACDB
ÀB
s
DC
Z2
=
ZI
Given
Altemate
angles
Now
BD
s
BD
AABD
=
ACDB
Z4
=
Z3
—
AD
BC
II
Common
S.
A.
S.
postulate
(i)
(corresponding
angles
of
congruent
triangles)
From
....(ii)
(i)
and
Also
ÀD
=
BC
(iii)
Corresponding
sides
of
congruent
As
ABU
DC
....(iv)
Given
Hence
ABCD
is
a
parallelogram
From
(ii)
-
(iv)
EXERCISE
11.2
(
1
)
Prove
that
a
quadrilatéral
(a)
is
a
parallelogram
congruent
if
its
Opposite
angles
are
(b)
Diagonals
bisect
each
other.
Given
ABCD
is
a
quadrilatéral.
mZA
=
mZC,
mZB
=
mZD
ABCD
is
a
parallelogram.
Broof
Statements
Reasons
Given
Given
Angles
of
a
quad.
mZA=mZC
mZB=mZD
Now
(i)
(ii)
mZA
+
mZB
+
mZC
+
mZD
=
360°
mZA
+
mZ
B
+
mZA
+
mZB
=
360°
mZA
+
mZA
+
mZB
+
mZB
=
360°
2mZA
+
2mZ
B
=
360°
(mZA
+
mZ
B)
=
360°
/2
=
ÂDIIBC
Similarly
it
1
From
(i),
(ii)
R^irranging
80°
Dividing
by
2
As
mZA
+
mZB
=
180°
can
be
1
!
(sum
of
interior
angles)
ÀB
CD
Hence
ABCD
is
a
parallelogram.
Proved
that
(2)
p
rove
that
a
quadrilatéral
In
quadrilatéral
is
a
parallelogram
if
its
opposite
sides
are
congruent.
„
D
ABCD,
ÂB
=DC,
C
ÂD
=BC
ABCD
is
a
gm
II
ÂBII
CD,
ÂDIIBC
Construction
loin
point
B
to
D
and
name
the
anales
Zl. Z2.
Z3
and
Z
4

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin