2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227 REAL NUMBERS TOPIC-1 RATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS Q.1 Identify a rational number among the following number: √ 25 6 , √ 20 4 , 2.27 , √2 , √3 [2.27 , 2014] Q.2 Express the rational number 0. 9 in the form , where p and q are integers and q ≠ 0. [1,2014] Q.3 Is √98 √2 is a rational number or not. [7,2012] Q4 write the simplest form of a rational number 177 413 . [3/7, 2012] Q.5 calculate the decimal which represent the fraction 7 8 [0.875,2012] Q.6 Identify the smallest number among the following numbers: 1.101001….., 1.1101001….,1.011012….., 1.011 . [1.011 = 1.011011011…., 2012] Q.7 What is the maximum number of digits in the repeating block of 1 , where n is a prime number. [ n-1, 2012] Q.8 Write a real number which has terminating decimal expansion. [0.248,2012] Q.9 Calculate the value of 2. 9 in the form of , where p and q are integers and q ≠ 0. [3, 2012] SHORT ANSWER TYPE QUESTIONS-I Q.1 Express 0.6 in the form , where p and q are integers and q ≠ 0. [2/3, 2013] Q.2 If 7x = 1, then find the decimal expansion of x. [x = 0.142857 , 2012] Q.3 Represent 0.237 in the form , where p and q are integers and q ≠ 0. [237, 2012] Q.4 Express 2157 625 in the decimal form and state whether it is terminating or not. [3.4512, 2012] Q.5 insert three rational numbers between 3 5 and 5 7 . [ 22/35,23/35,24/35, 2012] SHORT ANSWER TYPE QUESTIONS-II Q.1 Find three rational numbers between 5 7 and 9 11 . [386/539, 387/539,388/539, 2012] Q.2 Find six rational numbers between 3 and 4. [22/7,23/7, 24/7, 25/7, 26/7, 27/7 2014] LONG ANSWER TYPE QUESTIONS-I Q.1 Arrange in descending order √2 3 , √5 4 , √7 6 , and √3 12 Q.2 Express 0.32 + 0.35 in the form , where p and q are integers and q ≠ 0. [1659/990, 2012] TOPIC-2 IRRATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS Q.1 Identify the irrational number among the following numbers: 0.13, 0.1315 , 01315 , 0.3013001300013… Q.2 Is the product of two irrational numbers always an irrational number? [No, 2012] Q.3 Calculate the in the form of 0.777, where p and q are integers and q ≠ 0. [x=7/9, 2012] Q.4 Write the sum of √5 2 , & √7 3 . [ √5 2 + √7 3 , 2012] Q.5 Calculate the irrational number between 2 & 2.5. Q.6 Write the sum of 0. 3 And 0.4 . [x=7/9, 2012] Q.7 Simplify the decimal expansion of the number √2 . Q.8 Simplify is the number (√2 + √5 ) 2 . SHORT ANSWER TYPE QUESTIONS-I Q.1 Find any two irrational numbers between 0.1 and 0.12. [2014] Q.2 Find any two irrational numbers between 0.5 and 0.55. [2014] Q.3 Express the decimal numbers 2.218 in the form , where p and q are integers and q ≠ 0. [122/55, 2013] Q.4 Find any two irrational numbers between 1 7 and 2 7 , when it is given that 1 7 = 0. 142857 . [2012] SHORT ANSWER TYPE QUESTIONS-II Q.1 Represent √4.5 on the number line. [2014] Q.2 Represent √5 on the number line. [2014] Q.3 Express 0.6 in the form of , where p and q are integers and q ≠ 0. [2013]
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2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
REAL NUMBERS TOPIC-1
RATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS
Q.1 Identify a rational number among the following
number: √25
6, √
20
4, 2.27̅̅̅̅ , √2, √3 [2.27̅̅̅̅ , 2014]
Q.2 Express the rational number 0.9̅ in the form 𝑝
𝑞, where
p and q are integers and q ≠ 0. [1,2014]
Q.3 Is √98
√2 is a rational number or not. [7,2012]
Q4 write the simplest form of a rational number177
413.
[3/7, 2012]
Q.5 calculate the decimal which represent the fraction 7
8
[0.875,2012]
Q.6 Identify the smallest number among the following
numbers: 1.101001….., 1.1101001….,1.011012…..,
1.011̅̅ ̅̅ ̅. [1.011 = 1.011011011…., 2012]
Q.7 What is the maximum number of digits in the
repeating block of 1
𝑛, where n is a prime number.
[ n-1, 2012]
Q.8 Write a real number which has terminating decimal
expansion. [0.248,2012]
Q.9 Calculate the value of 2.9̅ in the form of 𝑝
𝑞, where p
and q are integers and q ≠ 0. [3, 2012]
SHORT ANSWER TYPE QUESTIONS-I
Q.1 Express 0.6̅ in the form 𝑝
𝑞, where p and q are integers
and q ≠ 0. [2/3, 2013]
Q.2 If 7x = 1, then find the decimal expansion of x. [x = 0.142857̅̅ ̅̅ ̅̅ ̅̅ ̅̅ , 2012]
Q.3 Represent 0.237̅̅ ̅̅ ̅ in the form 𝑝
𝑞, where p and q are
integers and q ≠ 0. [237, 2012]
Q.4 Express 2157
625 in the decimal form and state whether
it is terminating or not. [3.4512, 2012]
Q.5 insert three rational numbers between 3
5 and
5
7.
[ 22/35,23/35,24/35, 2012]
SHORT ANSWER TYPE QUESTIONS-II
Q.1 Find three rational numbers between 5
7 and
9
11.
[386/539, 387/539,388/539, 2012]
Q.2 Find six rational numbers between 3 and 4.
[22/7,23/7, 24/7, 25/7, 26/7, 27/7 2014] LONG ANSWER TYPE QUESTIONS-I
Q.1 Arrange in descending order √23
,√54
, √76
, and √312
Q.2 Express 0.32̅̅̅̅ + 0.35̅̅̅̅ in the form 𝑝
𝑞, where p and q are
integers and q ≠ 0. [1659/990, 2012]
TOPIC-2
IRRATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS
Q.1 Identify the irrational number among the following
Q.14 If x and y two positive real numbers such that 8x3 +
2y3 = 730 and 2x2y +3xy2 = 15, then evaluate 2x+3y. [10, 2012]
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
LINES & ANGLES TOPIC-1
Different Types of Angles
VERY SHORT ANSWER TYPE QUESTIONS
Q.1 In the fig. below, AOB is a straight line. Calculate
the measure of ∠COD.
D
C 2x-200
X+200 600
A B
Q.2 What is the measure of an angle which is
compliment of itself. [450]
Q.3 Write a complimentary angle of 650. [250]
Q.4 Two angles (300 – a) and (1250 + 2a). If each one is
the supplement of the other, then find the value of a.
[250]
Q.5 Write the complement of (900 – a). [a]
Q.6 Write the angle which is one fifth of its complement. [150]
Q.7 In the triangle, two angle, measure (550 + 3a) and
(1150 – 2a). If each is supplement of the other, then calculate the value of a. [100]
Q.8 In the given fig. what is the value of x? [500]
C
D
x X+100 X+200
A B
Q.9 Calculate the value of x in the figure below. [200]
5x 4x
A O B
Q.10 In the figure below, Calculate the value of y.
[280]
3y
400 2y A B
SHORT ANSWER TYPE QUESTIONS I Q.1 In the figure, lines XY and MN intersect at O. If
∠POY = 900 and a:b = 2:3, find the value of c. [1260]
P
M
a
b O
X Y
c
Q.2 If ∠ AOP = 5y, ∠ QOD = 2y and ∠BOC = 5y in the
given figure, find the value of y. [150]
A D
5y O 2y
P Q 5y
C B
Q.3 In the figure, if x + y = w + z, then prove that AOB
is a line C [1800] B
x
y w
z
A D
Q.4 In the given figure, ∠AOC and ∠ BOC form a line AB. If a-b = 800, find the value of a and b. [500]
C
A b A B
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q.5 In the given figure, ∠AOB: ∠BOC = 2:3. If ∠AOC
= 750, then find the measure of ∠AOB and ∠BOC. [450] A
B
O 750 C
Q.6 In figure, prove that , ∠AOB + ∠BOC + ∠COD +
∠DOA = 3600.
B A
O
D
C
Q.7 In figure ∠DOB = 870 and ∠COA = 820. If ∠BOA
= 350, then find ∠COB and ∠COD. [400]
C D
B 820
350 A
Q.8 In the figure, a is greater than b, by 1
6 of a straight
angle. Find the values of a and b. [750]
a b
Q.9 Two supplementary angles are in the ratio 2:3, find
the angles. [720 , 1080]
SHORT ANSWER TYPE QUESTIONS II
Q.1 In the given figure, PO Ʇ AB. If x:y:z = 1:3:5, then
find the degree measure of x,y and z.
[x=100, y=300,z=500] P
Q
R y x
z
A O B
Q.2 Prove that if two lines intersect each other, then the
bisectors of vertically opposite angles are in the same line.
Q.3 In figure, if AB || CF and CD || FE, then find the value of x. [750]
B
F x E 400
650
A C D
Q.4 I the given figure ∠3 and ∠4 are exterior angles of quadrilateral ABCD at point D and B respectively. And
∠A = ∠2, ∠C = ∠1. Prove that ∠3 + ∠4 = ∠1 + ∠2. D C
3 1
2 4
A B
Q.5 Lines PQ and RS intersect each others at O (see
figure). If ∠POR: ∠ROQ = 5:7, find all the angles a, b, c, and d. [1050]
p S
a
d O b c
R Q
Q.6 In the given figure, AC||DE and AD||CE, find x and
y, when it is given that ∠BAC = 700 and ∠DEC = 550. [550]
A x D
700
Y 550
B C E
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q.7 In figure, POQ is a line. Ray or is Ʇ to PQ. OS is
another ray lying between OP and OR. Prove that:
∠ROS = 1
2 [∠QOS - ∠POS]
R
S
P O Q
Q.8 It is given that ∠XYZ = 640 and XY is produced to a
point P. Draw a figure from the given information.
If ray YQ bisect ∠ZYP, find ∠XYQ and reflex ∠QYP.
[1220, 3020]
Q x Z x 640
P Y X
LONG ANSWER TYPE QUESTIONS
Q.1 In the given figure, two straight lines PQ and RS
intersect each other at O. If ∠POT = 700, find the value of a, b, c. [c=480]
Q
R
2c a
4b
750 b S
P T
TOPIC-2
Transversal Line VERY SHORT ANSWER TYPE QUESTIONS
Q.1 From the given figure, identify the incorrect statement, given that l || m and t is the transversal.
5 6
1 2 8 7 3
4
l m
Q.2 In fig. below, calculate the value of angle q.
P
A C
500 x 900
a y
D B
Q Q.3 In the given fig. AB and CD are parallel to each
other then calculate the value of x. [1000]
A B
1200
x E 1400
C D
Q.4 In the given figure, AB || CD and ‘l’ is transversal, then calculate the value of ‘x’. [260]
l
A 3x+350 B
y
2x+150
C D
Q.5 In fig., If m || n and ∠a:∠b = 2:3, then what will be
the measure of ∠h? [1080]
b a
m
n
h
Q.6 In the given fig. PQ || RS and EF || QS. If ∠PQS =
600, then what will be the measure of ∠REF? [1200]
P R
E F
600
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q S
Q.7 In fig. PQ || RS, ∠QPR = 700, ∠ROT = 200, find the
value of x. [500]
P Q 700
T
1200
O
SHORT ANSWER TYPE QUESTIONS-I
Q.1 In the fig. AB || DE, ∠ABC = 1400 and ∠CDE =
1000. Find ∠BCD. D E
A B 1000
1400
[600] C
Q.2 In the given fig. m || n and p || q. If ∠1 = 750 Prove
that ∠2 = ∠1 1
3 of right angle.
m
1
2 n
P q
Q.3 In the given fig., if l1 || l2 and l3 || l4, what is y in
term of x? [900 - 𝑥
2]
L3 l4
x l1
2
y l2
y
Q.4 In given fig. ΔLMN is an isosceles triangle with ∠m
= ∠n and LP bisects ∠NLQ. Prove that LP || MN.
Q
L
P
M N
Q.5 In the fig. AB || DC. Determine x. [2850]
A B
450
x O
300
D C
Q.6 In the given figure find x and y and then show that
l || m. t
500 l
x
y m
1300
Q.7 In the given figure, show that AB || EF. A
B
600
E F
1550
350
250
C D
Q.8 In the figure AB || CD, find the value of z, ∠DNM
and ∠CNM. [570]
M A B
(3z-420)
(2z+130) C N D
Q.9 In figure in AB || CD || EF and x:y = 3:2, find z.
[1080]
A B
x
A y B
A z B
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q.10 Find the supplement of 4
3 of the right angle. [600]
Q.11 In the given figure, state which lines are parallel and why.
B D A 700
G E 800
800 C H
F
Q.13 If a transversal intersects two parallel lines, then the bisectors of any pair of alternate angles are parallel.
Prove it.
A M B
l Q x
y P
C N D m
SHORT ANSWER TYPE QUESTIONS-I
Q.1 In figure, a transversal l cuts two lines AB and CD
at E and F respectively. EG is the bisector of ∠AEF and
FH is the bisector of ∠EFD such that ∠a = ∠b. Show that
EG || FH and AB || CD. [2014]
l
A E B
a
G H
b
C F D
Q.2 In figure, AB || CD, then find x. [1000]
A G B
1350 E x
1250 C D
Q.3 If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel,
then prove that the two lines are parallel. [2013]
M Q
1
A 2 B P
S
3 4
C R D
N
Q.4 In the figure, AB || CD, EF Ʇ CD and ∠GFC = 1300.
Find x, y and z. [500]
A E G B
y x
1300 z
C F D
Q.5 In the given figure, QP || ML, find the value of x.
[300]
Q A M
1350
150 100
x
P B L
Q.6 Prove that the angle between internal bisector of one
base angle of a triangle is equal to one-half of the
vertical angle. [2013] A E
B C D
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q.7 If two parallel lines are intersected by a transversal, prove that the bisector of the interior angles on the same
side of transversal intersect each other at right angles.
E [2012]
A M B
1
3 p
2
C N D
F
Q.8 In the given figure, AB || CD and EF is a transversal
cutting them at G and H respectively. If ∠EGB = 350 and
QP Ʇ EF, then find ∠PQH [2012]
E
A 350 B G
C Q H D
P
F
Q.9 In the given figure, if AB || CD, ∠BPQ = (5x-200) and ∠PQD – (2x-100), find the value of y and z. [2012]
[Y=500, z= 500]
A y B
P 5x-200
Q 2x-100
C z D
Q.10 In the figure, AB || CD, EF || DQ. Determine
∠PDQ, ∠AED and ∠DEF. [720]
P Q
430
C D
F
650
A E B Q.11 In the given figure l || m || n. From the figure, find
the ratio of ( x = y) : (y – x). [2012]
A B
l
1000 400
x0 E m
C D
y0
300 200 n
E
LONG ANSWER TYPE QUESTIONS
Q.1 In the given figure, AB || DC, ∠BDC = 350 and
∠BAD = 800. Find x, y and z. [350, 650, 1100]
D C
350 z
y
y-30
800
x
A B
Q.2 In the above figure ABCD is a quadrilateral in
which ∠ABC = 730, ∠C = 970 and ∠D = 1100. If AE ||
DC and BE || AD and AE intersect BC at F, find the
measure of ∠EBF. [270]
D C
1100 970
A F E
730
B
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
TOPIC-3
Angle Sum Property of triangle VERY SHORT ANSWER TYPE QUESTIONS
Q.1 What is the value of x in the figure given below? [400]
1100 600
Q.2 In the figure below, if x, y and z are exterior angles
of ΔABC, then calculate the value of x + y + z. [3600]
A
z
y B
x C
Q.3 In the figure below, if ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = k right angles, then what is the value of k? [k=4] A
F E
B C
D
Q.4 An exterior angle of a triangle is 800 and two interior opposite angles are equal. What will be the measure of
each? [400]
Q.5 In the given figure, calculate the value of ∠PQR. [300]
750 P
1050
Q R
Q.6 In ΔABC, ∠A = ∠B/2 = ∠c/6, then what will be the measure of ∠A? [200] Q.7 In the given figure, ABC is an isosceles triangle
with AB = AC and ∠A = 500. Calculate ∠B. [650]
A
500
B C
Q.8 In the fig. below, in ΔABC, AB = AC, then calculate
the value of x A [1300]
800
B C
x
Q.9 In the given figure below, what will be the value of
x? [1300]
A
1100
1200 B
x C
SHORT ANSWER TYPE QUESTIONS-I
Q.1 In the given figure, if ∠A = 600 and ∠B = 700, then
find∠ ACD. [1300]
A
600
700 1200
B C D
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q.2 In ΔABC, ∠A + ∠B = 650 and ∠B + ∠C = 1400, find the value of ∠B and ∠C. [B=250, C=1150] Q.3 In figure, if AB || CD, ∠APQ = 400 and ∠PRD = 1180, find x and y. [780] A B 400
y
x 1180
C Q R D
Q.4 In ΔABC, if ∠A = (2x-50), ∠B = (5x+50), ∠C =
(3x+500), then find the value of x, ∠A, ∠B and ∠C. [210, 700, 890]
Q.5 Prove that if one angle of a triangle is equal to the
sum of the other two angles, then the triangle is right
angled triangle. [2012]
Q.6 In ΔABC, ∠A = 600, ∠B = 400. Which side of this triangle is the smallest? Give reason for your answer? [2012] A
600
400
B C
Q.7 An exterior angle of a triangle is 1100 and one of the
interior opposite angles is 300. Find the measure of
another two angles of the triangle. [800]
Q.8 In ΔABC, ∠B = 450, ∠c = 550, AD bisects ∠A. Find ∠ADB and ∠ADC. [850] Q.9 Find the value of x in the given figure, where ∠A = 400 and ∠BED = 1200. [100]
A
400
E
F 1200
x B C D
Q.10 In the given figure AP and DP are bisectors of ∠A
and ∠D. Prove that 2∠APD = ∠B + ∠C. [2012]
A B
P
C D
Q.11 In figure, If lines PQ and RS intersect at point T,
such that ∠PRT = 500, ∠TSQ = 600 and ∠RPT = 1000,
find ∠SQT. [900] P
1000 500 T S
R 600
Q
Q.12 In the given figure AF || BE, AC Ʇ BE and AF
bisects ∠GAD. If ∠GAD = 700 then find the measure of
∠ABC and ∠ADE. [350]
G
A 700 F
B C D E
Q.13 Prove that if in a triangle, its sides are produced in
order, then sum of the exterior angles so formed is 3600.
[2012]
Q.14 The degree measure of three of a triangle are x0, y0,
and z0. If = x0+x0
2, then find the value of z0. [600]
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
SHORT ANSWER TYPE QUESTIONS-II
Q.1 In given figure DE Ʇ AB. Find the value of x
and y. [500, 600]
A
E
1100 F
x y 400
D C B
Q.2 If the figure QT Ʇ PR, ∠TQR = 400 & ∠SRP = 300 Find x and y. [800] P
300
T
400 y x
Q S R
Q.3 In figure, PQ Ʇ QR. QP || RL, ∠RQT = 380 and
∠QTL = 750. Find x and y. [530, 370]
P Q
y
380
x 750
R T L
Q4 In the given figure, ∠CAB : ∠BAD = 1:2, find all the internal angles of ΔABC. [2012] B
x
D
690 x+130
E A C
Q.5 In the given figure. BO and CO are bisectors of
∠DBC and ∠ECB respectively. If ∠BAC = 700 and
∠ABC = 400, find the measure of ∠BOC. [550] A
700
400
B C
?
D E O
Q.6 In the given figure, find a + b [1270]
B C
5( 𝑥
2-10)
1
2 x x + 90
A b D
a
Q.7 In figure AB || CD, then find the measure of x.
[180]
B D Q
x
P 1100 880
R A C
Q.8 In ΔABC, AD and CE are the bisectors of ∠A and
∠C respectively. If ∠ABC = 900 then find ∠AOC. [1350]
A
E
O
B D C
Q.9 In the given figure, AB || CD, ∠BAC = 720 and ∠CEF = 400. Find ∠CFE. [320]
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
E
D
400
B C
720 ?
A F Q.10 In the given figure, find the value of x0. [980]
A
230
D x0
400 350
Q.11 Prove that the sum of angles of a triangle is 1800
[2012]
Q.12 In the given figure, if ∠BCD = 250, ∠BAQ = 1100 and ∠ACR = 1250, then find the value of x, y and z. [550, 300, 800] Q 1100 A
D z
y 1250 x 250
B C R
Q.13 In the figure, ∠BAC = 500, ∠GBD = 700 and l and m are parallel lines. Find x, y and z. [1200, 700, 600] A
G 500 x
B C l 700
y z
D E m
Q.14 In figure, triangle ABC is right angled at A. AL is
drawn perpendicular to BC. Prove that ∠BAL = ∠ACB.
[2012] A
B C
L
LONG SHORT ANSWER TYPE QUESTIONS
Q.1 Prove that the sum of three angles of a triangle is 1800. Using this result, find the value of x and all the
three angles of the triangle, if the angles are:
(2x-7)0, (x+25)0 and (3x+12)0. [x = 250] A
B C
Q.2 Prove that the sum of all the angles of a triangle is
1800. Also find the angle of a triangle if they are in ratio 5:6:7.
Q3 The sides AB and AC of ΔABC are produced to
points P and Q respectively. If bisectors BO and CO of
∠CBP and ∠BCQ respectively meet a point o, then
prove that ∠BOC = 900 - 1
2 x.
Q.4 In the given figure, AB || CD, ∠ECD = 240, ∠EDC = 420 and AC = CE. Find x and y. [420, 720]
A B
x
y
E
z
C 240 420 D
Q.5 In the given figure, ∠ACD = ∠ABC and CP bisects ∠BCD. Prove that ∠APC = ∠ACP. A
D
P
B C
Q.6 In the given figure, find the value of x and y if AB ||
CD. [420, 380]
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
l
Q
A B
200 y
P x
220 C D
580 R
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
TRIANGLES TOPIC-1
Criteria for Congruence of
Triangle
VERY SHORT ANSWER TYPE QUESTIONS Q.1 In ΔABC and ΔDEF, AB = DE, ∠A = ∠D. What will be the condition in which the two triangles will be
congruent by SAS axiom?
Q.2 Given ΔOAP = ΔOBP in the figure given below.
Prove the criteria by which the triangles are congruent.
A
P
O
B
Q.3 In The figure given below, if AB = QR, BC = PR
and CA = PQ, then ……………………
A P
B C Q R
Q.4 In the given figure AD = BC and ∠BAD = ∠ABC, then prove that ∠ACB ≈ ∠BDA. D C
A B
Q.5 The exterior angle of a triangle is equal to the sum
of two……………….
Q.6 What do we call a triangle if the angle are in the ratio 5:3:7?
Q.7 In the figure below, it is given that ΔABC ≅ ΔBAC.
What criteria is used to prove that the triangles are congruent? D C
A B
Q8 In the given figure, if AB = DC, ∠ABD = ∠CBD,
which congruent rule would you apply to prove
ΔABD ≅ ΔCBD?
D C
A B
Q.9 Among the following which is not a criteria for
congruence of two triangles?
Q.10 In ΔAOC and ΔXZY, ∠A = ∠X, AO = XZ, AC = XY, then by which congruence rule ΔAOC ≅ ΔXZY ? A X
O C Y Z
Q.11 ΔABC ≅ ΔPQR, AB = PQ. Which statement has been followed in this?
SHORT ANSWER TYPE QUESTIONS-I
Q.1 In the figure ΔABC and ΔDBC are two isosceles
triangles on the same base BC. Prove that ∠ABD =
∠ACD. A
B C
D Q.2 Prove that the sum of the four angles of a
quadrilateral ABCD is 3600, using properties of
triangles. A B
6 1
5 2
3 7
4 8
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
C D
Q.3 In the figure below, ABC is a triangle in which AB = AC. X and Y are points on AB and AC such that AX =
AY. Prove that ΔABY ≅ ΔACX
A
X Y
B C
Q.4 In the figure below. ABCD is a square and P is the
mid-point of AD and CP are joined. Prove that ∠PCB =
∠PBC.
A B
P
D C
Q.5 In the figure below, O is the mid-point of AB and CD, prove that AC = BD.
A
D
O
C
B
Q.6 In the figure below the diagonal AC of quadrilateral
ABCD bisects ∠BAD and ∠BCD. Prove that BC = CD.
D
C
A
B
Q.7 In the figure, AO = OB and OD = OC. Show that:
(i) ΔAOD ≅ ΔBOC, (ii) AD || BC.
B
C
O
D
A
Q.8 In the given figure, D is the mid-point of base BC. DE and DF are perpendiculars of AB and AC
respectively such that DE = DF. Prove that ∠B = ∠C. A
E F
B C
Q.9 In figure ∠B = ∠E, BD = CE and ∠1 = ∠2. Show ΔABC ≅ ΔAED. A
B D C E
Q.10 In figure, AX = BY and AX || BY, prove that
ΔAPX ≅ ΔBPY
X B
P
A Y
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
SHORT ANSWER TYPE QUESTIONS-II
Q.1 In figure, PQRS is a square and SRT is an equilateral triangle. Prove that:
(i) PT = QT T
(ii) ∠TQR = 150
S R
P Q
Q.2 In the given figure AD = BC and BD = AC. Prove
that, ∠ADB = ∠BCA and ∠DAB = ∠CBA.
A B
D C
Q.3 In the given figure, if ∠ACD = ∠ACE and AB = BC, then prove that AE = CD.
B D A
x
E y
C
Q.4 In the given figure, AD is the bisector of ∠BAC and
∠CPD = ∠BPD. Prove that ΔCAP ≅ ΔBAP and CP = BP.
C
A P D
B
Q.5 ABCD is a quadrilateral in which AD = BC and
∠DAB = ∠CBA. Prove that BD = AC.
D
A
B
C
Q.6 In the given figure, prove that: CD + DA + AB + BC > 2AC.
C B
D A
Q.7 In the given figure, BA⊥CA, RP⊥QP, AB = PQ and
BR = CQ. Prove that AC = PR. A
R B Q C
P
Q.8 In the given figure, ABCD is a square and M is the
mid-point of AB. PQ⊥CM meets AD at P and CB produced at Q. Prove that PA = BQ.
D C
P
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
A M B
Q
Q.9 In figure, AB = EF, BC = ED, AB⊥BD, FE⊥EC.
Prove that ΔABD ≅ ΔFEC.
A F
B E
C D
LONG ANSWER TYPE QUESTIONS
Q.1 If D is the mid-point of the hypotenuse AC of a right
triangle ABC, prove that BD = 1/2AC.
A
E D
B C
Q.2 In the given figure, if AC = BC, ∠DCA = ∠ECB and
∠DCB = ∠EAC, then prove that BD = AE.
D E
A C B
Q.3 In figure ΔABC and ΔABD are such that AD = BC,
∠1 = ∠2 and ∠3 = ∠4. Prove that BD = AC.
D C
1 2
3 4
Q.4 Prove that two triangles are congruent if any two
angles and the included side of one triangle in equal to
any two angles and the included side of the other
triangle.
Q.5 If two parallel lines are intersected by a transversal,
then prove that bisectors of the interior angles from a rectangle.
Q.8 In the given figure, PQRS is a quadrilateral and T
and U are points on PS and RS respectively, such that
PQ = RQ, ∠PQT = ∠RQU and ∠TQS = ∠UQS. Prove that: QT = QU P T
S
Q
U R
Q.9 In the given figure, AM⊥BC and AN is the bisector
of ∠A. If ∠ABC = 700 and ∠ACB = 200, find the value
of ∠MAN. [250] A
700 200
B M N C
Q.10 In figure ABCD is a quadrilateral in which AD =
BC and ∠DAB = ∠CBA. Prove that:
(i) ΔABD ≅ ΔBAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC
A
D
B
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
C
Q.11ΔABC and ΔDBC are two isosceles triangles on the
same base BC and vertices A and D are on the same side
of BC. If AD is extended to intersect BC at P, show that:
(i) ΔABD ≅ ΔACD
(ii) ΔABP ≅ ΔACP
(iii) AP bisects ∠A as well as ∠D.
Q.12 In the figure BL⊥AC, MC⊥LN, AL = CN and BL
= CM. Prove that ΔABC ≅ ΔNML.
M
A L
C N
B
TOPIC-2
Some Properties of Triangle
SHORT ANSWER TYPE QUESTIONS
Q.1 Triangle ABC is an isosceles triangles such that AB
= AC. Side BA is produced to D, such that AD = AB.
Show that ∠BCD is a right angle.
Q.2 Prove that each angle of an equilateral triangle is
600.
Q.5 PQR is a triangle in which PQ = PQ. S is any point
on the side PQ. Through S, a line is drawn parallel to QR intersecting PR at T. Prove that PS = PT.
LONG SHORT ANSWER TYPE QUESTIONS Q.1 In figure ABCD is a square and EF is parallel to
diagonal BD and EM = FM. Prove that:
(i) DF = BE
(ii) (ii) AM bisects ∠BAD. [2012]
(iii)
D A
F
M
C E B
Q.2 In figure, it is given that RT = TS. ∠1 = 2∠2 and ∠4 = 2∠3. Prove that T
A B
1 4
2 3
R S
TOPIC-3
Inequalities of a Triangle
SHORT ANSWER TYPE QUESTIONS
Q.1 In ΔPQR, if S is any point on the side QR. Show that PQ + QR + RP > 2PS.
Q.2 In the given figure, PQR is a triangle and S is any point in its interior. Show that SQ + SR < PQ + PR.
P
S
Q R
Q.3 In ΔABC, if AB is the greatest side, then prove that
∠C > 600
LONG SHORT ANSWER TYPE QUESTIONS Q.1 Prove that the sum of two sides of a triangle is
greater than twice the median with respect to the third
side.
Q.2 ABCD is a quadrilateral in which AB and CD are
smallest and longest sides respectively. Prove that ∠A > ∠C and ∠B > ∠D.
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Q3 Show that the difference of any two sides of a
triangle is less than the third side.
2327 E SUDAMA NAGAR RING ROAD INDORE 452009 8109438041, 9893821227
Co- ordinate Geometry TOPIC-1
Cartesian System
SHORT ANSWER TYPE QUESTIONS-I
Q.1 Locate and write the co-ordinates of a point:
(A) Above x-axis lying in y-axis at a distance of 5 units from origin.
(B) Below x-axis lying in y-axis at a distance of 3 units
from origin. (C ) lying on x-axis to the right of origin at a distance of
5 units.
(D) Lying on x-axis to the left of origin at a distance of 2
units.
Q.2 In which quadrant do the following points lie?
(A) (-6, 2) (B) (-5, -4) (C ) (3, -2) (D) (9,6)
Q.3 Find the co-ordinates of the point which lies on y-
axis at a distance of 4 units in negative direction of y-axis.
(A) (-4, 0) (B) (4, 0)
(C ) (0, -4) (D) (0,4)
Q.4 In which quadrants do the following points lie?
(A) (2. -1) (B) (-1, 7) (C ) (-2, -3) (D) (4,5)
Q.5 In which quadrants will the points lie, if:
(A) The ordinate is 2 and the abscissa is -3? (B) the abscissa is -4 and the ordinate is -2?
(C ) The ordinate is 3 and the abscissa is 4?
(D) The ordinate is 3 and the abscissa is -2?
SHORT ANSWER TYPE QUESTIONS-II
Q.1 Plot the point A (1,3), B(1,-1), C(-1,-1). What must
be the coordinates of the point D, if ABCD is a rectangle?
Q.2 In which quadrilateral or on which axis does each of
the following points lie. (-5, 3), (4, -3), (5, 0), (6, 6), (-5, -4)?
Q.3 Find the co-ordinates of a point: (A) Which lies on x and y-axis, both.
(B) Whose abscissa is 5 and ordinate is 6.
(C ) Whose ordinate is 6 and which lies on y-axis.
(D) Whose ordinate is -4 and abscissa is -7. (E) Whose abscissa is 3 and which lies on x-axis.
(F) Whose abscissa is 4 and ordinate is 4.
Q.4 Plot the following points and write the name of the figure, thus obtained. A (3,0), B(5,0), C(5,3) and D(3,3).
Q.5 Write the coordinates of the vertices of a rectangle whose length and breadth are 6 and 3 units respectively,
one vertex at the origin, the longer side lies on the x-axis
and one of the vertices lies in the III quadrant.
LONG ANSWER TYPE QUESTIONS
Q.1 If the coordinates of a point M are (-2,9) which can
also be expressed as (1 + x, y2) and y > 0, then find in
which quadrant do the following points lie:
P(y,x), Q(2,x), R(x2,y-1), S(2x,-3y). [2012]
Q.2 Observed the points plotted in the figure and find the
following: [2012] (i) The co-ordinates of E
(ii) The point with the co-ordinates (-4, -1).
(iii)The abscissa of A – abscissa of B
(iv)The ordinates of C + ordinate of F.
Q.3 Plot (-3.0), (5,0) and (0,4) on Cartesian plane. Name
the figure formed by joining these points and find its area. [16 sq. units, 2012]
Q.4 In which quadrant or on which axis do each of the points (-2,4), (3,-1), (-1,0), (-3,-5), and (1,2) lie?
Verify your answer by locating them on the Cartesian
plane.
Q.5 Plot the following points in the Cartesian plane: