CLASS 9 TRIANGLES
TRIANGLE
Competency Focused Questions (MCQs)
Q1. The measure of each of the base angles of an isosceles triangle whose
base angle is double the vertex angle is:
(a) 58° (b) 64° (c) 72° (d) 80°
Q2. PQR is a right triangle in which ∠Q = 90°. If ∠P : ∠R = 2 : 3, then
measure of least angle is:
(a) 36° (b) 54° (c) 56° (d) 18°
Q3. In figure, if QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°, the value of y - x is:
(a) 80° (b) 50° (c) 30° (d) 130°
Q4. In a ΔABC, if ∠A - ∠B = 42° and ∠B - ∠C = 21°, then ∠B = ?
(a) 32° (b) 63° (c) 53° (d) 95°
Q5. In the given figure, CB and BA of ΔABC have been produced to D and E
respectively such that ∠ABD = 110° and ∠CAE = 135°. Then, ∠ACB = ?
(a) 65° (b) 45° (c) 55° (d) 35°
Q6. The sides BC, CA and AB of ΔABC have been produced to D, E and F
respectively, then ∠BAE + ∠CBF + ∠ACD = ?
(a) 240° (b) 300° (c) 320° (d) 360°
Q7. In the given figure, two rays BD and CE intersect at a point A. The side
BC of have been produced on both sides to points F and G respectively.
If ∠ABF = x, ∠ACG= y and ∠DAE = z then z = ?
(a) x + y - 180° (b) x + y + 180°
(c) 180° - (x + y) (d) x + y + 360°
Q8. In the given figure, side BC of ΔABC has been produced to a point D. If
∠A ∠B = x, ∠ACB = 5y and ∠ACD = 7y. Then, the value of x is:
(a) 60° (b) 50° (c) 45° (d) 35°
CBE I(b). Select Response Questions (MCQs)
Q9. Which of the following statement/ statements is/are true for any
triangle?
(i) All sides are equal.
(ii) It has exactly two acute angles.
(iii) The sum of the angles is always 180°.
(iv) The longest side is always twice the shortest side.
Choose the correct option from the following:
(a) (i) and (ii) (b) (i) and (iv)
(c) Only (iii) (d) (iii) and (iv)
Q 10. A triangle that has one angle greater than 90° is called:
(i) Equilateral triangle (ii) Acute triangle
(iii) Right triangle (iv) Obtuse triangle
Q 11. In the given figure, OB and OC are the angle bisectors of ∠ABC and
∠ACB respectively.
Which of the following statements are true?
(i) The value of x + y is 40°.
(ii) The value of m is 100°.
(iii) The value of x + y is 50°.
(iv) The value of m is 80°
Choose the correct option from the following:
(a) Only (1) (b) Only (ii)
(c) (iii) and (iv) (d) Only (iii)
CBE II. Short Answer Questions (Constructed Response Questions)
Q 12. In a ΔABC, ∠A - ∠B = 33° and ∠B –∠C = 18°. Find the angles of the
triangle.
Q 13. A ΔABC is right angled at A and L is a point on
side BC such that AL ⊥ BC. Prove that ∠BAL
= ∠ACB.
Q 14. The side BC of ΔABC is produced to D. The
Bisector of ∠A meets BC at E. Prove that
∠ABC + ∠ACD = 2∠AEC.
Q 15. In the adjoining figure, show that
∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 360°.
Q 16. In a ΔABC, it is given that ∠A : ∠B : ∠C = 3: 2 : 1
andCD ⊥ AC. Find ∠ECD.
Q 17. In the given figure, AB || CD and EF is a
transversal, cutting them at G and H
respectively. If ∠EGB = 35° and QP ⊥ EF, find
the measure of ∠PQH.
18. In the adjoining figure, AB = AD, ∠BAP = ∠DAQ and ∠PAC = ∠QAC.
Prove that AP = AQ.
CBE I(a). Competency Focused Questions (MCQs)
19. In the triangle ABC, AD is the bisector of ∠A, and AB = AC. Which option correctly completes the statement given below? By _______________ congruency criteria, ΔABD ≅ ΔACD and using CPCT, we get ∠ABD = _______________.
(a) ASA; ∠ADB (b) SAS; ∠ACD
(c) ASA; ∠ADC (d) SAS; ∠ADC
20. In the given figure, if AD is the median, then ∠BAD is:
(a) 32° (b) 38° (c) 49° (d) 55°
Q21. In the adjoining figure, AB || BE and FE || BE.
If AB = FE and BC = DE, then
(a) ABD ~ EFC
(c) ABD ~ ECE
(b) ABD ~ FEC
(d) ABD ~ CEF
Q22. In the adjoining figure, AC = BD. If
∠CAB = ∠DBA, then ∠ACB is equal to
(a) ∠RAD (b) ∠ABC
(c) ∠ABD (d) ∠BDA
Q23. In ∆ABC and ∆PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles
are:
(a) isosceles but not congruent
(b) isosceles and congruent
(c) congruent but isosceles
(d) neither congruent nor isosceles
Q24. In triangles ABC and DFE, AB = FD and ∠A = ∠D. The two triangles
will be congruent by SAS axiom if:
(a) BC = EF
(b) AC = DE
(c) AC = EF
(d) BC = DE
Q25. In the given figure, the measurement of x is:
(a) 15° (b) 20°
(c) 30° (d) 40°
Q26. In the given figure, the value of a + b is:
(a) 110° (b) 120°
(c) 130° (d) 150°
Q27. In the given figure, AB = BC and
AC = CD. Then ∠BAD : ∠ADB =
(a) 1 : 1 (b) 3 : 1
(c) 1 : 3 (d) 1 : 2
Q28. In ∆ABC, ∠A = 40°, AB = AC.
Then ∠B : ∠C =
(a) 1 : 1 (b) 1 : 2
(c) 2 : 1 (d) 1 : 3
CBE I(b). Select Response Questions (MCQs)
Q29. Two triangles are shown. The perimeter of ∆PUL is 30 cm. Are the
triangles congruent?
(i) Yes, as on calculating the missing angle in each triangle it can be
concluded that the triangles are
congruent by AAA criteria.
(ii) No, as the missing angle in each
triangle cannot be calculated.
(iii) Conclusion about the
congruency of triangles can be
made provided the length
of the side NQ of triangle MNQ is known.
(iv) Conclusion about the congruency of
triangles can be made provided the length
of the side MN or side MQ of triangle
MNQ is known.
Choose the correct option from the following:
(a) (i) and (iv) (b) (ii) and (iii)
(c) (iii) and (iv) (d) None of the above
Q30. In ∆ABC and ∆PQR, AB = PQ, ∠A = ∠P and ∠B = ∠Q, which of the
following criteria can be used to prove that ∆ABC ≅ ∆PQR ?
(i) SAS (Side Angle Side) (ii) AAS (Angle Angle Side)
(iii) ASA (Angle Side Angle) (iv) AAA (Angle Angle Angle)
Choose the correct option from the following:
(a) (i) and (iv) (b) (ii) and (iii)
(c) (iii) and (iv) (d) None of the above
Q31. If ∆PQR ≅ ∆XYZ and ∆PQR is not congruent to ∆ZXY. Then which of
the following is are not true?
(i) QR = XY (ii) PR = XZ
(iii) PQ = XY (iv) YZ = QR
Choose the correct option from the following:
(a) Only (ii) (b) (ii) and (iv)
(c) Only (i) (d) (iii) and (iv)
Q32. In ∆ABC and ∆PQR, ∠B = ∠Q, AB = PQ and BC = QR. Which criteria
can be used to prove that ∆ABC and ∆PQR are congruent?
(i) SAS (ii) ASA (iii) AAS (iv) AAA
Choose the correct option from the following:
(a) (i) and (ii) (b) Only (i)
(c) (iii) and (iv) (d) (i), (ii), (iii) and (iv)
Q33. In the given figure, if PQ = TU, QR = SU and ∠PQS = 60° = ∠TUR.
Then which of the following statement/statements is/are true?
(i) ∆PQS ≅ ∆TRU (By SAS)
(ii) ∆PQS ≅ ∆RTU (By ASA)
(iii) ∆PQS ≅ ∆TUR (By SAS)
(iv) ∆PQS ≅ ∆UTR (By ASA)
Choose the correct option from the following:
(a) (i) and (ii) (b) (i) and (iii)
(c) (i) and (iv) (d) Only (ii)
II. Competency Focused & Inference Based Questions (A-R)
The following questions are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Q34. Assertion (A): In ΔABD and ΔACD, given
AD = AD, BD = CD
∠ADB = ∠ADC
∴ ΔABD ≅ ΔACD [By SAS congruency rule]
⇒ AB = AC
Reason (R): Corresponding parts of congruent triangles are equal.
Q35. Assertion (A): In quadrilateral ACBD, AC = AD and
AB bisects ∠A, then ΔABC ≅ ΔABD.
Reason (R): ΔABC ≅ ΔABD, by AAS congruence
rule.
Q36. Assertion (A): In two triangles ABC and PQR, AB = PQ, BC = QR and
∠B = ∠Q, then ΔABC ≅ ΔPQR.
Reason (R): Two triangles are congruent if two sides and one angle
of a triangle are equal to corresponding two sides and
one angle of other triangle.
Q37. Assertion (A): In ΔABC, AM ⊥ BC such that BM = CM, then ∠A = ∠B.
Reason (R): Two triangles are congruent if two sides and the
included angle of one triangle are equal to
corresponding two sides and the included
angle of other triangle.
Q38. Assertion (A): If two triangles are congruent to each other, then the
ratio of the corresponding sides is 1: 1.
Reason (R): Two triangles are congruent if and only if they have
same shape and size.
Q39. Assertion (A): If ΔABC ≅ ΔRPQ, then BC = QR.
Reason (R): Corresponding parts of two congruent triangles are
equal.
III. Short Answer Questions (Constructed Response Questions)
Q40. ABC is an isosceles triangle in which AC = BC. AD and BE are
respectively two altitudes to sides BC and AC. Prove that AE = BD.
[NCERT Exemplar]
Q41. O is a point in the interior of a square ABCD such that COD is an
equilateral triangle. Show that AOB is an isosceles triangle.
[NCERT Exemplar]
Q42. ABC is a right triangle such that AB = AC and bisector of angle C
intersects the side at D. Prove that AC + AD = BC.
Q43. l and m are two parallel lines intersected by another pair of parallel
lines p and q (see figure). Show that ΔABC ≅ ΔCDA.
[NCERT Exemplar]
Q44. The picture below shows a staircase outside a house. Each step of the
staircase's congruent and there are 25 steps in the staircase from the
floor to the platform and 25 steps from the platform to the roof.
What is the length of the staircase railing? [CFQ by CBSE]
Q45. In the given figure, if AB = BC and ∠A = ∠C, then find the value of x.
Q46. In the given figure, diagonal PR of quadrilateral PQRS bisects the
angles P and R. Prove that PQ = PS and RQ = RS.
Q47. ABC is an isosceles triangle with AB = AC, in which altitudes BE and
CF are drawn to equal sides AC and AB respectively. Show that these
altitudes are equal.
Q48. In the given figure, ∠QSM = ∠RSM and PM bisects ∠RPQ.
Q 62. The angles of a triangle are in the ratio 3 : 5 : 7. The smallest angle of
the triangle is:
(a) 12° (b) 36° (c) 60° (d) 84°
[CFQ by CBSE]
Q 63. ΔABC is an isosceles triangle with AB = AC. If the vertex angle is twice
the sum of the base angles, then the vertex angle of the triangle is:
(a) 30° (b) 120° (c) 60° (d) none of these
Q 64. In ΔABC, AB = AC and ∠B = 50°, then ∠C is equal to:
(a) 40° (b) 50° (c) 80° (d) 130°
Q 65. In ΔPQR, ∠R = ∠P, QR = 4 cm and PR = 5 cm. Then the length of PQ
is:
(a) 4 cm (b) 5 cm (c) 2 cm (d) 2.5 cm
Q 66. In the given figure, if the measure of
exterior angle ACD is x, then the value
of x is:
(a) 100° (b) 135°
(c) 140° (d) 150°
Q 67. In figure, AB = AC, CH = CB and HK || BC.
If ∠CAX = 137°, then ∠CHK equals:
(a) 68.5° (b) 43°
(c) 137° (d) 58.5°
Q 68. In the given figure, ∠BAC = 79°, CA = CB
and BD = CD. The measures of x, y and z
respectively are:
(a) 25°, 130°, 25°
(b) 45°, 90°, 45°
(c) 54°, 72°, 54°
(d) 22°, 136°, 22°
Q 69. In figure, AB = AC, ∠ACM = 125° and
∠PAB = x. The value of x is:
(a) 130° (b) 110°
(c) 100° (d) 120°
Q 70. In the figure shown, G is a point on PR and QG = QR. Which option
shows the correct steps to find the relationship between ∠QPR and
∠QRP?
(a) Step-1: ∠QRG = ∠QGR
Step-2: ∠QPG + ∠PQG = ∠QGR
Step-3: ∠QPG + ∠PQG = ∠QRG
Step-4: ∠QPR < ∠QRP
(b) Step-1: ∠RQG = ∠QGR
Step-2: ∠QPG + ∠PQG = ∠QGR
Step-3: ∠QPG + ∠PQG = ∠RQG ⇒ ∠QPG < ∠RQG
Step-4: ∠QPR < ∠QRP
(c) Step-1: As QG = QR, RQ < PQ
Step-2: ∠QPG > ∠RQG
Step-4: ∠QPR < ∠QRP
(d) Step-1: As QG = QR, ∠QRG = ∠QGR
Step-2: ∠QPG + ∠PQG = ∠QGR
Step-3: ∠QPG + ∠PQG = ∠QRG ⇒ ∠QPG < ∠QRG
Step-4: ∠QPR > ∠QRP
Q 71. P is a point on the bisector of ∠ABC. If the line through P, parallel to
BA meets BC Q, prove that BPQ is an isosceles triangle.
Q 72. In the given figure, ∠ABD is an exterior
angle of ΔABC.
(i) Find the value of x.
(ii) Find the measure of ∠ABC.
[CFQ by CBSE]
Q 73. In ΔABC, AD is the perpendicular bisector of BC. Show that ΔABC is
isosceles in which AB = AC.
Q 80. In the adjoining figure, ABCD is a square
and P is mid-point of AD, BP and CP are
joined. Prove that ∠PBC = ∠PCB.
Q 81. In the adjoining figure, find the value of x.
Q 82. In ΔABC, AB = AC and D is a point on AB such that AD = DC = BC.
Show that ∠BAC = 36°.
Q 83. ABCD is a square and ABE is an equilateral triangle outside the
square, prove that ∠ACE = 1/2 ∠ABE.
CBE V. Long Answer Questions (Constructed Response Questions)
Q 84. (a)In the figure (i), bisectors of ∠B and ∠C of an isosceles triangle
ABC with AB = AC intersect each other at O. BO is produced to a
point M. Prove that ∠MOC = ∠ABC.
[NCERT Exemplar]
(b) In the figure (ii), bisectors of ∠B and ∠C of an isosceles triangle
ABC with AB = AC intersect each other at O. Show that the
external angle adjacent to ∠ABC is equal to ∠BOC.
[NCERT Exemplar]
Q1. In the given figure, lines AB and CD intersect at a point O. The sides
CA and OB have been produced to E and F respectively such that
ZDAE = x and ZDBF = y.
If ZOCA = 80°, ZCOA = 40° and ZBDO = 70°, then x + y = ?
(a) 190° (b) 230° (c) 210° (d) 270°
Q2. In a AABC, it is given that ZA: ZB:ZC= 3:2:1 and ZACD = 90°. If BC is
extended to E, then ZECD = ?
(a) 60° (b) 50° (c) 40° (d) 25°
Q3. In the given figure, x and y are:
(a) x = 70°, y = 37° (b) x = 37°, y = 70°
(c) x = 47°, y = 60° (d) x = 60°, y = 47°
Q 4. In the given figure, BD ⊥ AC, the measure of ∠ABC is:
(a) 60° (b) 30° (c) 45° (d) 90°
Q 5. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA.
If ∠CAB = 30°, then the measure of ∠AOB is:
(a) 80° (b) 100° (c) 120° (d) 135°
Q 6. It is given that ∆ABC ≅ ∆FDE in which AB = 5 cm, ∠B = 40° and ∠A =
80°, then which of the following is true?
(a) ∠D = 60° (b) ∠E = 60°
(c) ∠F = 60° (d) ∠D = 8P
Q 7. If the angles of a triangle are (x - 40°), (x - 20°) and ( x/2 -10°), then
find the value of x. Give your answer in degrees.
Q 8. If AB = QR, BC = PR and CA = PQ, then ∆CBA is congruent to _______.
Q 9. In ∆ABC, AD is the perpendicular bisector of BC (see figure). Show
that ∆ABC is an isosceles triangle in which AB = AC.
[NCERT]
Q 10. In the given figure, AC = BC, ∠DCA = ∠ECB and ∠DBC = ∠EAC.
Prove that triangles DBC and EAC are congruent, and hence show
that DC = EC.
OR
In the given figure, BD || EF, ∠AEF = 55° and ∠ACB = 25°, find
∠ABC.
Q 11. In the figure below, the bisectors of angles B and C of a triangle ABC
intersect each other at the point D and ∠A = 50°. Find the value of
∠BDC.
OR
In the given figure, AD and BQ are
straight lines. BP = BC and DQ ||
CP. If ∠ABP = 4x and ∠CPD = x,
prove that
(i) CP = CD. (ii) DP bisects ∠CDQ.
Q 12. In a toy game, a robot starts from Home, picks an object from the
Shop, delivers it to the client and goes back Home.
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