triangles for ix

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

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ASSIGNMENT ON TRIANGLES

PART – A

Directions for 1 – 15: State true/false

1. Two lines are always congruent.

2. Two line segments are congruent if they have the same length.

3. Two circles are congruent if they have the same radius.

4. Two angles are congruent if the sum of their measures is equal to 180o.

5. If the three angles of one triangle are equal to the angles of another triangle, then the triangles are

similar.

6. If two triangles are congruent, then any angle of the first triangle will be equal to any angle of the second

triangle.

7. The diagonal of a quadrilateral divides it into two congruent triangles.

8. The diagonal of a trapezium divides it into two congruent triangles.

9. Two right triangles are always congruent.

10. All the three angles in an isosceles triangle are equal.

11. If two medians of a triangle are equal then the triangle must be isosceles.

12. If the perpendicular bisector of the side BC of ∆ABC passes through the vertex A, then AB = AC.

13. Two altitudes corresponding to the two equal sides of an isosceles triangle are not always equal.

14. The sum of two sides of a triangle can never be equal to the third side.

15. Any point on the perpendicular bisector of a line segment will be equidistant from the end points of the

line segment.

16. In the figure AD = BC and ∠DAB = ∠CBA. Prove that AC = BD.

17. AB is a line segment and ‘ℓ’ is its perpendicular bisector.

If P is any point on ‘ℓ’, prove that PA = PB.

P

B A

ℓ

A

B

C

D


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18. In the figure ℓ and m are parallel lines. These are intersected by another set of parallel lines ‘p’ and ‘q’.

Show that ∆ABD ≅ ∆CDB.

19. In the figure, AD is the bisector of ∠BAC and ‘P’ is a point on AD such that ∠CPD = ∠BPD.

Prove that CP = BP.

20. In a parallelogram ABCD, the two diagonals AC and BD are equal. Find the measure of ∠ABC.

21. In the figure, it is given that AB = CD, BE = EC, AB ⊥ BC and DC ⊥ BC. Prove that ∆ABE ≅ ∆DCE.

22. In the figure AB = CD and AD = BC. Prove that ∆ADC ≅ ∆CBA.

23. In the figure OP is the bisector of ∠AOB. Show that every point on OP is equidistant from the arms of

the angle.

A

B C

D

m

ℓ

q p

A P

C

B

D

A

B E C

D

A

B

E

C

D

P

A

O

B


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3

24. If angles A, B and C of a triangle ABC are equal to each other, prove that ∆ABC is equilateral.

25. In the figure, PQR is a triangle and QM and RN are altitudes. If QM = RN, show that ∠PQR = ∠PRQ.

26. If the angle bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is

isosceles.

27. Two sides AB, BC and median AM of ∆ABC are respectively equal to the sides PQ, QR and median PS

of another triangle PQR. Show that ∆ABC ≅ ∆PQR and ∆ABM ≅ ∆PQS.

28. Show that in a right triangle, the hypotenuse is the largest side.

29. In the figure, D is a point on the side BC of ∆ABC such that AB = AD. Show that AC > AD.

30. In the figure, AC > AB and AD is the bisector of ∠BAC. Show that ∠ADC > ∠ADB.

31. The diagonals AC and BD of quadrilateral ABCD intersect each other at O. Show that

AB + BC + CD + DA > AC + BD.

P

M N

Q R

D

A

B C

A

B D C

O

D

A B

C


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4

32. If ‘D’ is a point in the interior of ∆ABC, show that DB + DC < AB + AC.

33. In the figure, AB = AC. Show that AB < AD.

34. In the figure, AP is the shortest line segment that can be drawn from point ‘A’ to the line ‘ l ’.

If PR < PQ, show that AR < AQ.

35. In the figure, ‘D’ is a point on the side BC of triangle ABC. E is a point such that ED = BD. Show that

AB + AC > BE.

36. Show that the sum of the three altitudes of a triangle is less than the perimeter of the triangle.

37. In the figure, AB = AC. D is any point on AC. Show that BD > CD.

D

B

A

C

A

P Q R l

C

A

B D 50°

25°

A

B

D

C

B D

A

C

E


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5

38. In the figure AB >AC. The bisectors of ∠B and ∠C meet at O. Show that OC < OB.

39. In the figure, show that AB + BC + CD + DA > 2BD

A

B

O

C

A

D

C

B


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6

PART – B

Directions for 1 – 10: State true/false

1. Two figures are congruent, if they have the same shape.

2. Two circles of the same circumference are congruent.

3. Two squares of equal sides are congruent.

4. Two rectangles of equal lengths are congruent.

5. Two trapeziums with their corresponding non–parallel sides equal are congruent.

6. Two right triangles with equal hypotenuses are congruent.

7. Two isosceles triangles with equal bases are congruent.

8. Two squares with equal areas are congruent.

9. Two rectangles with equal areas are congruent.

10. Two circles with proportional radii are congruent.

11. If ∆ABC ≅ ∆EGF, which of the following statements is not always correct?

(A) ∠BAC = ∠GEF (B) ∠ACB = ∠EFG (C) AC = EF (D) AB + BC = GE + EF

12. Equal parts of the following triangles are marked with identical signs.

Which of the following relationships is correct?

(A) ∆ABC ≅ ∆PQR (B) ∆ACB ≅ ∆PQR (C) ∆BAC ≅ ∆QPR (D) ∆CAB ≅ ∆PQR

13. Which of the following is not a congruency criterion?

(A) SSS (B) SSA (C) ASA (D) SAS

14. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that BD = AC.

R

P

Q C

A

B

A

B

C

D


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7

15. In the figure AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.

16. In the figure AP and BQ are perpendicular to AB. If AP = BQ,

Show that ‘O’ is the mid point of AB and PQ.

17. In the figure, m || n and M is the mid points of AB. Prove that M is also the mid point of CD.

18. In the figure, the line segment joining the mid–points of the sides AB and CD is perpendicular to both

the sides. Prove that AD = BC.

19. In the figure, AB || CD and P is the mid point of BD. Prove that ∆CPD ≅ ∆APB.

D

A

P

O

Q

B

A C

B D

M

m

n

A M B

C N D

A B

P

C D

A

E

C B


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8

20. In the figure, altitude AD bisects BC. Show that ABC is an isosceles triangle.

21. In the figure, altitudes BD and CE are equal. Prove that ∆ABC is isosceles.

22. In the following figures, AC = PR, BC = QR and ∠ABD = ∠PRS. Prove that ∠P = ∠A.

23. ABC is an isosceles triangle with AB = AC. BE and CF are altitudes from B and C, respectively. Show

that BE = CF.

24. ABC and DBC are two isosceles triangles with AB = AC and BD = DC, respectively, on the same base

BC. Show that ∠ABD = ∠ACD.

A

E F

B C

A

C B

D

A

D E

B C

A

C B D

P

R Q S

A

C B D


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9

25. In the figure, ABC is an isosceles triangle with AB = AC. Side BA is produced to D such that AD = AB.

Show that ∠BCD is a right angle.

26. ABC is a right triangle right angled at A and AB = AC. Find the measures of ∠C and ∠B.

27. Show that the angles of an equilateral triangle are 60o each.

28. In the figure, ABC and DBC are two triangles on the same base BC, such that AB = AC and DB = DC,

prove that ∠ABD = ∠ACD.

A

B C

D

C

A B

A

D

B C


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10

29. In the figure, AE is the bisector of the external angle CAD such that AE || BC. Show that ABC is an

isosceles triangle.

30. In the figure, ∠A = 90o. D is the mid point of the hypotenuse BC and DM ⊥ BA. Prove that AB = AC.

31. In the figures, the two sides and one median, namely AB, BC and AM of ∆ABC are respectively equal to

the two sides and the median PQ, QR and PN of the other triangle ∆PQR. Show that

(i) ∆ABM ≅ ∆PQN (ii) ∆ABC ≅ ∆PQR

32. In the figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that BE = ED.

A

D

E

C B

A L

C

D M

B

A

M C B

P

N R Q

A

D B

C

E


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11

33. In the figure, AD ⊥ CD and BC ⊥ CD. If AQ = BP and DP = CQ, prove that ∠DAQ = ∠CBP.

34. ABCD is a square. X and Y are points on the sides AD and BC, respectively, such that AY = BX. Prove

that BY = AX and ∠BAY = ∠ABX.

35. In the figure, Q is a point on side SR of ∆PSR such that PQ = PR. Prove that PS > PQ.

36. In the figure, ‘S’ is a point on the side QR of ∆PQR. Prove that PQ + QR + RP > 2PS

37. Prove that the sum of the three sides of a triangle is greater than the sum of its three medians.

38. In the figure, T is a point on side QR of ∆PQR and S is a point such that RT = ST.

Prove that PQ + PR > QS.

A

D P Q C

B

A B

Y X

D C

S Q

R

P

Q S

R

P

P R

Q

T

S


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12

39. In the figure, AC > AB and D is a point on the AC such that AB = CD. Prove that CD < BC.

40. In the figure, PQ > PR, QS and RS are the bisectors of ∠Q and ∠R, respectively. Show that SQ > SR

41. In the figure, PQ = PR. Show that PS > PQ.

42. In the figure, AD is the bisector of ∠A. Show that AB > BD.

43. ABC is a triangle. Locate a point in the interior of ∆ABC which is equidistant from all the vertices of the

triangle ABC.

44. In ∆ABC, ∠B = 40o, ∠C = 60o and the bisector of ∠BAC intersects BC at X Arrange the segments AX,

BX and CX in the descending order of their lengths

A

D

C B

P

R Q

S

B D

C

A

P

R Q

S

A

C X

B 60o

40o


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13

45. If ‘D’ is the mid–point of the hypotenuse AC of a right angled ∆ABC. Prove that BD = 21

AC.

46. In the figure, prove that OA + OB + OC + OD > AC + BD

47. In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the

smallest side.

48. In the figure, XY is a diameter and ‘O” is the centre of the circle. Z is a point on the circle.

Prove that XY > XZ.

49. In a triangle ABC, AC > AB, the bisector of ∠A meets BC at D. Prove that ∠ADB is acute.

A

B

C

O

D

X

O

Y

Z


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14

ANSWER KEY

TRIANGLES

PART – A

1. True 2. True 3. True 4. False

5. False 6. False 7. False 8. False

9. False 10. False 11. False 12. True

13. False 14. True 15. True 20. 900

PART – B

1. False 2. True 3. True 4. False

5. False 6. False 7. False 8. True

9. False 10. False 11. (D) 12. (B)

13. (B) 44. . AX = BX > CX

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