class 9 geometry

Revision

Monday, July 6, 2020 8:45 AM

Geometry = geometry
Triagle :- Triangle is a closed figure having three angle . Triangle

According to sides According to angle
Equilateral ( having all sides equal &each angle 60) Acute ange traingle (all angles are less than 90 ie
Isosceles triangle ( two sides equal ) acute angle )
Scalene triangle ( having non of them equal) Right angle triangle ( having one angle 90 others
are acute)
Obtuse ange traingle ( having one angle obtuse ie
greater than 90 less than 180 and reaming angle
are acute

Sum of complementary angle is 90
Sum of supplementary angle is 180
Vertically opposite angle ( same vertex but opposite direction) are equal{ shape (X)}.
Alternate angle are equal and shape is (N or Z)
Corresponding angle are equal and shape is F
Co-interior angle' sum is 180 and shape is U or C
Axiom :An axiom is self evidence truth
Postulate: Geometrical assumption without proof
There are 5 axiom of congruency

1) SSS 2) RHS 3)ASA 4) SAS 5)AAS or SAA

2) But no axiom of SSA

3) Symbol of congruency is
4) Note: Remaining sides or angle are called corresponding sides or angle
5) All congruent triangle are similar but all similar triangle are not congruent

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Theorem 6:10 PM

Saturday, August 8, 2020

Prove that sum of angles tringle is 180

Given :- PQR is a tringle .
To Prove :- .
Construction :- through the vertex P, draw AC parallel to QR

Statement Reason

1. Straight angle

2. Being alternate angles in

3. Being alternate angles in

4. Whole part axiom

5. From statement (1),2,3,&4

So, sum of angles tringle is 180

Theorem 2

The exterior angle of triangle is equal to sum of two opposite
interior angles

Given:-

Construction :- Through the vertex C, draw CE

Statement Reason

1) Whole part axiom

2) Being alternate angles in

3) Being corresponding angles in

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3) Being corresponding angles in
4) Adding statement 2&3
5) From statement 1,2,3,4
Thus, the exteriior angle of atriangle is equal to the sum
of two opposite interior angles

Theorem 4
Base anlage of isosceles triangle are equal

Given :- Reason
To prove :-
Construction :- by construction
Statement by given
1) In being common side
i) by RHS axiom
Ii) AB=AC corresponding angles of congruent triangle
Ii) AD= AD
2)
3)

Hence ,Base angle of isosceles triangle are equal

Converse of theorem 4
If two angle of a triangle are equal then the sides opposite to them are also equal
Given : In
To prove :
Construction :

Statement Reason

1) In

i) Given

Ii) by Construction

Ii) AD= AD common side

2) by AAS axion

3) coreesponding sides of congurent traingle

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Theorem 5

The bisector of the vertical angle of an isosceles triangle

is perpendicular to the base and bisect the base

Given : in isosceles , AB=AC,AD is the bisector of

To prove :

Statement Reason
1) In
i) Given
Ii) AD is the bisector o
Ii) AD= AD common sides
2) by SAS axiom
3) corresponding sides of congruent triangle
corresponding
4) Being linear pair equal in statement 4

5)

Converse of theorem 5

The line joining the vertex and the mid point of the base

of an isosceles triangle is perpendicular to the base and

bisect the vertical angle

Given : in isosceles , AB=AC &D is the mid point

of BC

To prove :

Statement Reason
1) In
i) D is the mid point BC
Ii) AB=AC Given
Ii) AD= AD common side
2) by sss axiom
1) corresponding angle of congurent traingle
corresponding angles of congurent traingle
4
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4 corresponding angles of congurent traingle
5 being linear pair equal

from statement 5

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Tringle re

Saturday, June 11, 2022 12:02 PM

1a) from the given information find the values of x

equal to sum of two opposite interior angle

1c)

equal to sum of two opposite interior angle

1f)

Ii) equal to sum of two opposite interior angle
correspondding angles
iii)
Iv) x=64

1h)

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1h) equal to sum of two opposite interior angle

Solution
i)
Ii)

or

Solution

QR is produce to S

I)

II)

55+b=180

b=180-55

b=125 xterior angle is equal to sum of two opposite interior anle

III)

2a) in the each of the following figure find the value of the x

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Solution
CG // AB//FE is drawn

i) a+135=180
a=180-135
a=45
Ii) b+142 =180
b=38
Iii)
x= 45+38
x=83
2c)

Solution
EF//AB//DC is drawn
i) a=55
ii) b=65
iii)
x=55+65
x=120

3) From each of the following figure find the value of the x

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3) From each of the following figure find the value of the x

solution
1)

Or
2)

130=x+55
x=130-55
X=75

3c)

solution

1)
Or

2)
110=x+40

3e)

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Solution
1)
2)
or
3h)

Solution
1)
2)

115=x+75

Solution
1)

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1)
2)

90=x+75
x=90-75
x=15

4a)

4a)
From first and third
A = 2C &

4b)

4d)

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R=Q-21= 53-21=32
P=Q+42
5a) The sum of two angles of a triangle is 116 and their difference is 24 . Find the measurement of each of the angles

Solution
Let two angles are x and y then remaining is z

sum of angles of triangle is 180

116+z=180
z= 180-116
=64
Difference is 24 so x-y=24
x=y+24
from I
x+x+24=116
or 2x=116-24
or 2x= 92
or

4c) if sides of triangle are produce in order , prove that the sum of exterior angle so
formed is equal to four right angle .
Solution

From figure
from figure

Now ,

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Now ,
6a) in a

Given :-BO is the bisector of & CO is the bisector of
To find :- value of
1

3

Or

6c)

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6c)

1) 65+30+A= 180 sum of angle of triangle
A=180-95
A =85

2) whole part axiom
Or
or being AN is the bisector of the
or
x= 42.5
3) 90+a+65=180
or a= 180-155
a= 25
4) whole part axiom

7) in the given figure, the side BC is produce to D if BE is the bisector of
the bisector of the

1)
2)
3) from statement 1&2

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Isosceles triangle

Saturday, June 11, 2022 12:00 PM

2a)
is an isosceles tringle . If AB=AC and BAC=70 , find

solution
base angle of isoscelse triangle

3a) from the information given below figure the value of x and y

Solution

i) a=65
Ii)
Ii)

x
2d)

Solution

i) 50=c
Ii)

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x=100
3a)

Solution

I) --> base angle of isosceles trianlge
Ii) --> sum of angles of

Iii) being alternate angle
b=50
base angle of isosceles tringle
iv) sum of angles of traingle
I)

x=80 then prove that :
4a)
in the figure PQ=PR . If PS bisect

Solution

i) a=b
Ii) x=y

5b)

5 Geometry Page 2

Exterior angle of trainle s equla to sum of two opposite interior anlge
b=35+35=70

70=c B=40, and AE=AD then
Exterior anlge is equla to

y=35+70
y=105
Q 6 a)

in the given figure ,EC is the bisector of
find the measurement of

Solution

Solution :-

b=70

c= 70

sum of angles of triangle

70+70+

7a) in the given figure ,QS and RS are the bisectors of if

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1) to sum of two opposite interior anlge

2) to sum of two opposite interior anlge

3)
Q 8) In the given figure ,MN=OP and PN=OM prove that :OQ=NQ

Given :- ,MN=OP and PN=OM
To prove :- OQ=NQ

Statement Reason

1) In

i) Given

Ii) ON=ON Common side

Ii) MO= PN given

2) By SSS axiom

3) Corresponding angles of congruent triangle

4)OQ=NQ if sides are equal then angle opposite to them are
equal

9a)

in an isosceles triangle , if the vertex angle is twice the sum of the base angle , calculate the angles
of triangle

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Solution ,

he vertex angle angle is twice the sum o f the base angle

10 a) In the given figure , o is the midpoint of each line segment AB and CD prove that AC=BD and
AC

Given :- o is the mid point of each line segment AB and CD
To prove :- AC=BD and AC

Statement Reason

1)

i) AO=OB o is the mid point of AB

Ii) Vertically opposite angle

Ii)CO=DO o is the mid point of CD

2) by SAS axiom

3) AC=BD corresponding sides of

4) Corresponding angles of congruent triangle

5)AC being alternate angle equal

Proved

Q 11 in the given figure AB=AC and OB=OC. Prove that

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Given : AB=AC and OB=OC Reason
in base angle of isosceles triangle
To prove : In
Subtracting statement 2 from 1
Statement Whole part axiom
1)
prove that AD=BE.
3
4

Q 12 )

In

Given :AB=AC, AC=CE and BD=BC
To prove : AD=BE

Statement Reason

1) base angle of isosceles triangle

2) In

i) given

Ii) =BCE Supplements of equal angle of statement 1

Ii) BD= BC Given

by SAS axiom

corresponding sides of congruent triangle

12 b) and are equilateral triangle s .Prove that AD=BE.

in the figure

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in the figure and are equilateral triangle s .Prove that AD=BE.

Given :- and are equilateral triangle s

To prove :- AD=BE. Reason
each angle of equilateral triangle
Statement both are equal to 60+

1 sides equilateral triangle ABC
from statement 2
2) sides equilateral triangle CDE
by SAS axiom
3) In corresponding angles of congruent triangle

i)

Ii)

Ii) CD= CE

5) AD=BE.
Proved

Q 13

Given : AB=AC , BX=CY and KX//AY Sn Reason
To prove : being AB=AC and base angle of isosceles triangle
Sn Statements corresponding angle
1 from statements 1 and 2
2 from statements 3 if two angles are equal then sides opposite to
3 them are equal
4 XK=XB Given
From statement 4 and 5
5 XB =CY
6 XK=CY

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6 XK=CY From statement 4 and 5
7
From statement 6
XK=CY supplements of equal angle from statements 2
vertically opposite angles
8) by SAA axiom

Given :
Q 14

Given : PQ=PR , Sn Reasons
to prove : QM=RN
Sn Statements PQ=PR
1 common side
i by AAS axiom
ii corresponding sides of the triangle
iii QR=QR
2
3 QM=RN
proved

15 )

Given : OA=OB, OC=OD Sn Reasons
to prove: AB//CD
Sn Statements 1 being AO=OB and base angle of isosceles triangle
1
2 2 being CO=OD and base angle of isosceles triangle
3
In exterior angle of triangle is equal to sum of two
4 opposite interior angles

exterior angle of triangle is equal to sum of two
opposite interior angles

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5 2x=2y from statement 4 and 3
Being alternate angle equal
6 AB//CD
Proved
16)

Given : CE is the bisector of ACB , BE=EC SN Reasons
To prove : CE is the bisector of ACB
Sn Statements BE=EC and base angle of isosceles triangle
1 From 1 and 2
2 Exterior angle is equal to sum of two opposite interior angle
3 from 3 and 4
4 Exterior angle is equal to sum of two opposite interior angle
5 from statement 5 and 6
6
7

proved

17)

Given: Sn Reasons
To prove : AC=BD being
Sn Statements
1 CE=ED Adding statements 1 and 2
2 EB=AE
3 CE+EB=ED+AE Common side

Ie CB=AD
4
i AB=AB

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i AB=AB Common side
ii given
iii CB=AD from statement
5 by SAS
6 AC=BD corresponding sides of congruent triangle

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