Triangles

Triangles
CLASS IX
Questions From CBSE Examination Papers
1. In ∆
ABC, ∠
C = ∠
A and BC = 6 cm and
AC = 5 cm. Then the length of AB is :
(a) 6 cm (b) 5 cm
(c) 3 cm
(d) 2.5 cm
10. In triangles ABC and DEF, AB = DE, BC = EF
and ∠
A= ∠
D. Are the triangles congruent ? If
yes, by which congruency rule ?
2. In ∆
PQR, ∠
P = 60°, ∠
Q = 50°. Which side of
the triangle is the longest ?
(a) PQ
(b) QR
(c) PR
(d) none
(a) yes, by SAS
(b) no
(c) yes, by SSS
(d) yes, by RHS
11. In the figure, if OA = OB, OD = OC, then ∆
AOD
3. In ∆
ABC and ∆
PQR, AB = PR and ∠
A= ∠
P.
∆
AOD ? ∆
BOC by congurence rule :
The two triangles will be congruent by SAS axiom
C
B
if :
(a) BC = QR
(b) AC = PQ
(c) AC = QR
(d) BC = PR
O
4. In ∆
ABC, ∠
A = 50°, ∠
B = 60°, arranging the
sides of the triangle in ascending order, we get :
(a) AB < BC < CA
(b) CA < AB < BC
(c) BC < CA < AB
5.
(d) BC < AB < CA
∆
ABC ? ∆
PQR. If AB = 5 cm, ∠
B = 40° and
∠
A = 80°, then which of the following is true ?
(a) QP = 5 cm, ∠
P = 60°
(b) QP = 5 cm, ∠
R = 60°
(c) QR = 5 cm, ∠
R = 80°
(d) QR = 5 cm, ∠
Q = 40°
6. If E is a point on side QR of ∆
PQR such that PE
bisects ∠
PQR, then :
(a) QE = ER
(b) QP > QE
(c) QE > QP
(d) ER > RP
7. In ∆
ABC if AB = BC, then :
(a) ∠
B>∠
C
(b) ∠
A= ∠
C
(c) ∠
A= ∠
B
(d) ∠
A < RP
8. If ∆
ABC ? ∆
DEF by SSS congruence rule
then :
(a) AB = EF, BC = FD, CA = DE
(b) AB = FD, BC = DE, CA = EF
(c) AB = DE, BC = EF, CA = FD
(d) AB = DE, BC = EF, ∠
C=∠
F
9. For the given triangle PQR, which of the following
is true ?
125°
P
A
(a) SSS
(b) ASA
D
(c) SAS
(d) RHS
12. In right triangle DEF, if ∠
E = 90°, then :
(a) DF is the shortest side
(b) DF is the longest side
(c) EF is the longest side
(d) DE is the longest side
13. If one angle of a triangle is equal to the sum of
the other two angles, then the triangle is :
(a) an isosceles triangle
(b) an obtuse angled triangle
(c) an equilateral triangle
(d) a right triangle
14. Two equilateral triangles are congurent when :
(a) their angles are equal
(b) their sides are equal
(c) their sides are proportional
(d) their areas are proportional
15.
∆
ABC ? ∆
PQR. If AB = 5 cm, ∠
B = 40° and
∠
A = 80°, then which of the following is true?
(a) QP = 5 cm, ∠
P = 60°
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(b) QP = 5 cm, ∠
R = 60°
(c) QR = 5 cm, ∠
R = 60°
Q
R
(d) QR = 5 cm, ∠
Q = 40°
100°
(a) PQ = QR
(c) PQ < QR
(b) PQ > QR
(d) ∠
P< ∠
Q
16. One of the angles of a triangle is 75°. If the
difference of the other two angles is 35°, then the
largest angle of the triangle has a measure of :
(a) 80°
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(b) 75°
(c) 100°
(d) 135°
17. In the figure, if AB = AC and AP = AQ, then by
which congruence criterion ∆
PBC ? ∆
QCB?
22. In the figure, which of the following statements is
true ?
A
16cm
B
(a) SSS
(b) ASA
(c) SAS
(a)
(b)
(c)
(d)
(d) RHS
15cm
C
19cm
∠
B=∠
C
∠
B is the greatest angle in triangle
∠
B is the smallest angle in triangle
∠
A is the smallest angle in triangle
23. It is not possible to construct a triangle when its
sides are :
(a) 8.3 cm, 3.4 cm, 6.1 cm
(b) 5.4 cm, 2.3 cm, 3.1 cm
(c) 6 cm, 7 cm, 10 cm
(d) 3 cm, 5 cm, 5 cm
18. In the figure, in ∆
ABC, AB = AC. The value of
x is :
A
80°
ABC ? ∆
24. If ∆
PQR, then which of the following is
true ?
C
B
(a) AB = RP
x
(a) 80°
(b) 100°
(c) 130°
(b) CA = RP
(c) AC = RQ
(d) CB = QP
25. In the given figure, AD is the median, then ∠
BAD
is :
(d) 120°
19. Given ∆
OAP? ∆
OBP in the figure. The criteria
by which the triangles are congruent is :
A
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A
P
O
40°
B
(a) 55°
B
(a) SAS
(b) SSS
(c) RHS
(d) ASA
20. In the figure, ABCD is a quadrilateral in which
AB = BC and AD = DC. Measure of ∠
BCD is :
C
D
(b) 50°
(c) 100°
(d) 40°
26. In the figure, PR = QR, ∠
PRA = ∠
QRB and
∠
BPR = ∠
AQR. Prove that BP = QA.
A
B
A
B
108°
42°
D
(b) 30°
(c) 105°
(d) 72°
A
21. In ∆
AOC and ∆
XYZ,
A= ∠
X, AO = XZ,
AC = XY, then by which congruence rule
∆
AOC ? ∆
XZY?
(a) SAS
(b) ASA
(c) SSS
(d) RHS
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R
27. In the figure, ∆
ABD and ∆
BCD are isosceles
triangle on the same base BD.
Prove that ∠
ABC =
∠
ADC.
C
(a) 150°
P
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B
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D
C
28. In the fgure, ACBD is a quadrilateral with
AC = AD and AB bisects∠
A. Show that
∆
ABC? ∆
ABD.
What can you say about BC and BD ?
38.
In ∆
ABC, AB = AC. D is a point inside ∆
ABC such
that BD = DC. Prove that ∠
ABD = ∠
ACD.
A
C
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A
D
B
B
39.
D
29. In a ∆
ABC, if AB = AC, ∠
A = 100°, then fnd
∠
B and ∠
C.
C
In the figure, X and Y are two points on equal sides
AB and AC of a ∆
ABC such that AX = AY.
Prove that XC = YB.
30. Prove that each angle of an equilateral angle is
60°.
A
31. In ∆
ABC, AD is perpendicular bisector of BC.
Show that ? ABC is an isosceles triangle in which
AB = AC.
X
32. Prove that in an isosceles triangle, angles opposite
to equal sides are equal.
33. In ∆
ABC, ∠
A = 60°, ∠
B = 40°. Which side of
this triangle is the smallest? Give reasons for your
answer.
Y
B
40.
C
In the figure, D is the mid-point of base BC, DE and
DF are perpendiculars to AB and AC respectively
such that DE = DF. Prove that ∠
B=∠
C.
34. In the figure, AX = BY and AX||BY, prove that
∆
APX ? ∆
BPY.
A
B
X
P
F
E
A
Y
B
35. PS is an altitude of an isosceles triangle PQR in
41.
which PQ = PR. Show that PS bisects ∠
P.
36. In a ∆
DEF, if ∠
D = 30°, ∠
E = 60°, then which
side of the triangle is longest and which side is
shortest ?
37. In the figure, ∠
ABD = ∠
ACE and AB = AC.
Prove that ∆
ACE.
ABD ? ∆
C
D
In the figure, ABCD is a square and P is the
midpoint of AD. BP and CP are joined. Prove that
∠
PCB = ∠
PBC.
A
B
P
C
D
42.
In the figure, the diagonal AC of quadrilateral
ABCD bisects ∠
BAD and ∠
BCD.
Prove that BC = CD.
D
C
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A
B
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43.
In the figure, ∠
B<∠
A and ∠
C<∠
D. Show that
AD < BC.
D
B
C = 90°, M is midpoint of
53. In right triangle ABC, ∠
hypotenuse AB. C is joined to M and produced to
a point D such that DM = CM. Point D is joined
to point B. Show that :
(I) ∆
AMC ? ∆
BMD
(ii)∠
DBC = ∠
ACB
PQR.
54. In the figure, PR > PQ and PS bisects ∠
Prove that ∠
PSR > ∠
PSQ.
O
A
P
C
44.
Prove that an equilateral triangle can be constructed
on any given line segment.
45.
In the figure, AB > AC,
BO and CO are the
bisectors of ∠
B and ∠
C
respectively. Show that
OB > OC.
46.
47.
A
Q
R
S
O
55. AD is an altitude of an isosceles triangle ABC in
which AB = AC. Show that AD is also the median
of the triangle.
B
C
ABC and ∆
DBC are two isosceles
56. In the figure, ∆
triangles on the same base BC and the vertices A
Two sides AB and BC and median AM of ∆
ABC
and D are on the same side of BC. AD is extended
are respectively equal to the sides PQ and QR
to meet BC at P. Prove that AP bisects∠
A.
and median PN of ∆
QR.
Show that ∆
ABM ? ∆
PQN.
A
In the figure, D is any point on the base BC
produced of an isosceles triangle ∆
ABC.
Prove that AD > AB.
A
D
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B
B
D
C
48.
In an isosceles triangle ABC with AB = AC, BD
and CE are two medians. Prove that BD = CE.
49.
In the figure, if PS = PR, ∠
TPS = ∠
QPR, then
prove that PT = PQ.
T
R
S
57. In the figure, the perpendiculars AD, BE and CF
drawn from the vertices A, B and C respectively
of ∆
ABC are equal. Prove that the triangle is an
equilateral triangle.
A
F
P
50.
In ∆
ABC, BD and CE are two altitudes such that
BD = CE. Prove that ∆
ABC is isosceles.
51.
If ∆
ABC is an isosceles triangle with AB = AC,
prove that the perpendiculars from the vertices B
and C to their opposite sides are equal.
D is a point on side BC of ∆
ABC such that
AD = AC. Show that AB > AD.
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C
D
58. Prove that medians of an equilateral triangle are
equal.
59. In the figure, line segment AB||CD and O is the
mid point of AD. Show that :
(I) ∆
AOB ? ∆
DOC.
(ii) O is also the mid point of BC.
C
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D
O
A
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E
Q
B
52.
C
P
B
60.
In the figure, AD and BC are equal and
perpendiculars to the same line segment AB. Show
that CD bisects AB.
66.
In the figure, ∠
B=∠
C and AB = AC. Prove that
BE = CF.
67.
In the figure, ∆
ABC and ∆
DBC are two isosceles
triangles on the same base BC and vertices A and
D are on the same side of BC. If AD is produced
to intersect BC at P, then show that AP is the
perpendicular bisector of BC.
C
B
O
A
D
61.
In the figure, BD and CE are two altitudes of
∆
ABC such that BD = CE. Prove that ∆
ABC is
isosceles.
A
A
E
D
D
B
62.
63.
B
C
Show that in a right angled triangle the hypotenuse
is the longest side.
In the figure, if ∠
a > ∠
b, then prove that
PQ > PR.
68.
C
P
In the figure, ∆
LMN is an isosceles triangle with
LM = LN and LP bisects ∠
NLQ. Prove that
LP||MN.
Q
P
L
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N
M
64. In the figure, in ∆
ABC, D is any point in the interior
of ∆
ABC such that ∠
DBC = ∠
DCB and AB = AC.
Prove that AD bisects ∠
BAC.
69.
A
In the figure, ∠
QPR = ∠
PQR and M and N are
points respectively on sides QR and PR of ∆
PQR
such that QM = PN. Prove that OP = OQ; where
O is the point of intersection of PM and QN.
R
D
N
M
O
B
65.
C
Q
P
In the figure, D and E are points on the base BC
of a ∆
ABC such that BD = CE and AD = AE.
Prove that ∆
ABE ? ∆
ACD.
70.
In the figure, ∆
ABC is an isosceles triangle in which
AB = AC, side BA is produced to D such that AD
AD = AB. Show that ∠
BCD is a right angle.
D
A
A
B
B
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D
E
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C
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C
71.
In the figure, D is a point on side BC of ∆
ABC
such that AD = AC. Show that AB > AD.
77. In the figure, show that
2(AC + BD) > (AB + BC + CD + DA)
D
C
A
O
B
A
B
72.
C
D
In the ∆
ABC, BE and CF are altitudes on the sides
AC and AB respectively such that BE = CF. Using
RHS congruency rule, prove that AB = AC.
B
D
O
E
y
C
73.
79. O is a point in the interior of ∆
PQR. Prove that
1
OP + OQ + OR >
(PQ + QR + PR).
2
80.
x
A
78. In the figure, if ∠
x =∠
y and AB = AC, then prove
that AD = AE.
In the figure, AC = AE, AB = AD and∠
BAD =
∠
EAC. Show that BC = DE.
In an isosceles triangle ABC with AB = AC, the
bisector of ∠
B and ∠
C intersect each other at O.
Join A to O. Show that :
(i) OB = OC
(ii) AO bisects ∠
A
81. Show that perimeter of a triangle is greater than
the sum of its medians.
82.
In the figure, ∠
BCD = ∠
ADC and ∠
ACB = ∠
BDA.
Prove that AD = BC and ∠
A =∠
B.
A
74.
In the figure, AB is a line segment. P and Q are
points on opposite sides of AB such that each of
them is equidistant from the points A and B. Show
that the line PQ is perpendicular bisector of AB.
P
B
D
C
83. AB and CD are respectively the smallest and the
longest sides of a quadrilateral ABCD as shown in
the figure. Prove that ∠
A> ∠
C and ∠
B>∠
D.
D
A
C
B
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A
C
B
Q
75.
In ∆
’s ABC and PQR, AB = PQ, AC = PR and
altitude AM and altitude PN are equal. Show that
∆
ABC ? ∆
PQR.
76.
Prove that the sum of any two sides of a triangle
is greater than twice the length of median drawn
to the third side.
84. In the figure, if two isosceles triangles have a
common base, prove that the line segment joining
their vertices bisects the common base at right
angles.
A
B
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D
C
85.
In the figure, AD is a median of ∆
ABC and BL
and CM are perpendiculars drawn from B and C
on AD and AD produced respectively. Prove that
BL = CM.
92.
In the figure, ∠
QPR = ∠
PQR and M and N are
respectively points on sides QR and PR of ∆
PQR,
such that QM = PN. Prove that OP = OQ, where
O is the point of intersection of PM and QN.
A
R
L
B
N
C
D
M
O
M
Q
P
86.
87.
88.
Prove that two triangles are congruent if any two
93.
angles and the included side of one triangle is
equal to any two angles and the included side of
the other triangle.
In a triangle ABC, AB = AC, E is the mid point
of AB and F is the mid point of AC. Show that
BF = CE.
In the figure, AC = BC, ∠
DCA = ∠
ECB and
∠
DBC = ∠
EAC. Prove that (i) ∆
DBC ? ∆
EAC;
(ii) DC = EC and BD = AE.
D
In the figure, ABCD is a square and ∆
DEC is an
equilateral triangle. Prove that :
(i) ∆
ADE ? ∆
BCE, (ii) AE = BE, (iii) ∠
DAE = 15°.
E
E
D
C
A
B
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A
89.
90.
B
C
In ∆
ABC, AB = AC, and the bisectors of ∠
B and
∠
C intersect at point O. Prove that BO = CO and 94.
the ray AO is the bisector of ∠
BAC.
In the figure, S is any point in the interior of ∆
PQR.
Show that SQ + SR < PQ + PR.
P
In the figure, BA||PQ, CA||RS and BP = RC. Prove
that (i) BS = PQ; (ii) RS = CQ.
S
A
S
Q
R
Q
95.
In the figure, PQ and RS are perpendicular to QS,
QA = BS and PB = AR.
Prove that ∠
QPB =∠
SRA.
R
P
B
91.
C
R
P
In a right triangle ABC, right angled at C, M is the
mid point of hypotenuse AB. C is joined to M and
produced to a point D such that DM = CM. Point
D is joined to point B. Show that :
(I) ∆
AMC ? ∆
BMD (ii) ∠
DBC = 90°
(iii) ∆
DBC ? ∆
ACB
D
Q
96.
B
A
A
P
D
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S
In figure, AP and DP are bisectors of two adjacent
angles A and D of a quadrilateral ABCD. Prove
that 2∠
APD = ∠
B+ ∠
C.
M
B
B
A
C
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C

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