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Assignments in Mathematics Class IX (Term I)

7. TRIANGLES

IMpoRTANT TERMS, DEfINITIoNS AND RESuLTS

l Figures having the same shape and size are called congruent figures.

l Two circles of the same radii are congruent. l Two squares of the same sides are congruent. l If two triangles ABC and DEF are congruent

under the correspondance A → D, B → E and C → F, then symbolically it is expressed as ∆ABC ≅ ∆DEF.

l Two triangles are congruent, if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. It is called SAS congruence rule.

l Two triangles are congruent, if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. It is called ASA congruence rule.

l Two triangles are congruent, if any two pairs of angles and one pair of corresponding sides are equal. It is called AAS congruence rule.

l Angles opposite to equal sides of an isosceles triangle are equal.

l The sides opposite to equal angles of a triangle are equal.

l Each angle of an equilateral triangle is 60°. l If three sides of one triangle are equal to the three

sides of another triangle, then the two triangles are congruent. It is called SSS congruence rule.

l If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. It is called RHS congruence rule.

l If two sides of a triangle are unequal, the angle opposite to the longer side is larger.

l In any triangle, the side opposite to the larger angle is longer.

l The sum of any two sides of a triangle is greater than the third side.

1. In the figure, if AD is an altitude of an isosceles triangle ABC in which AB = AC, then :

. (a) ∠BAD = ∠CAD (b) ∠BAD = ∠ABD (c) ∠BAD = ∠ACD (d) none of these 2. ∆ABC and ∆ACD in the figure are congruent by

which criterion?

(a) SAS (b) SSS (c) ASA (d) none of these

3. In ∆ABC, if AB = AC and ∠B = 70°, then ∠A is equal to :

(a) 70° (b) 40° (c) 65° (d) none of these

4. In a ∆ABC, if ∠A = 45° and ∠C = 60°, then the shortest side of the triangle is :

(a) AB (b) AC (c) BC (d) none of these

5. Which of the following figures may not be congruent ?

(a) two squares of equal sides (b) two line segments of equal lengths (c) two circles of equal radii (d) two triangles of equal angles

6. If two sides of a triangle are unequal, then : (a) the longer side has smaller angle opposite to it (b) the smaller side has greater angle opposite to it

Summative aSSeSSment

Multiple ChoiCe Questions [1 Mark]

a. important Questions

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(c) the longer side has greater angle opposite to it. (d) none of these

7. In the figure, AB = AD and BC = DC. If ∠BAC = 30°, the measure of ∠OAD is :

(a) 60° (b) 45° (c) 30° (d) 40°

8. In the figure, if AD = BC and AD BC, then :

(a) AB = AD (b) AB = DC (c) BC = CD (d) none of these

9. In the given figure, if AC = AD and AB bisects ∠A, then :

(a) BC = BD (b) BC = AB (c) BC = AC (d) none of these

10. In the given figure, ∆ABC ≅ ∆DEF by :

(a) SAS congruence rule (b) ASA congruence rule (c) SSS congruence rule (d) RHS congruence rule

11. If ∆ABD and ∆ACD are congruent by SSS congruence rule, then :

(a) ∆ABC is an equilateral triangle (b) ∆BDC is an equilateral triangle (c) ∠B = ∠C (d) ∠DBC = ∠ACD

12. In ∆ABC, if AB = AC and ∠A = 80°, then the values of ∠B and ∠C respectively are :

(a) 50°, 60° (b) 50°, 50° (c) 40°, 40° (d) 80°, 50°

13. For a ∆ABC, which of the following statements is true ?

(a) AB + BC + AC = 0 (b) AC – AB > BC (c) AC – AB < BC (d) none of these

14. If AD and BE are altitudes of a ∆ABC and AE = BD, then :

(a) AD > BE (b) AD < BE (c) AE = BE (d) AD = BE

15. Which of the following is not a criterion for congruence of triangles ?

(a) SAS (b) ASA (c) SSA (d) SSS

16. In ∆ABC and ∆PQR, if AB = QR, BC = PR and CA = PQ, then

(a) ∆ABC ≅ ∆PQR (b) ∆CBA ≅ ∆PRQ

(c) ∆BAC ≅ ∆RPQ (d) ∆PQR ≅ ∆BCA

17. In ∆ABC, BC = AB and ∠B = 80º. Then ∠A is equal to :

(a) 80° (b) 40° (c) 50° (d) 100°

18. In ∆PQR, ∠R = ∠P and 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

19. D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then :

(a) BD = CD (b) BA > BD (c) BD > BA (d) CD > CA

20. In the given triangle, AD = AC, ∠ACD = 75°

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and ∠BAD = 35°. The longest side of the ∆ABC is :

(a) AB (b) BC (c) AC (d) all are equal

21. Read the two statements and choose the correct option :

Statement P: All equilateral triangles are isosceles triangles.

Statement Q: All scalene triangles are isosceles triangles.

(a) P is true but Q is false (b) P is false but Q is true (c) Both P and Q are true (d) Both P and Q are false

22. In the given figure, if AB = AC and DB = DC, the ratio of ∠ABD to ∠ACD is :

(a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) none of these

23. In the figure, if AB = CD and AD = BC, then :

(a) ∆ADC ≅ ∆CAB (b) ∆ADC ≅ ∆BAC (c) ∆ADC ≅ ∆CBA (d) none of these

24. In ∆ABC, side AB is produced to D so that BD = BC. If ∠ABC = 60° and ∠A = 70°, then :

(a) AD > CD (b) AD > AC (c) AD < CD (d) both (a) and (b)

25. In the figure, if AB||DC and ∠B = ∠D, then :

(a) ∆ABC ≅ ∆ADC (b) ∆ABC ≅ ∆CDA (c) ∆ABC ≅ ∆DCA (d) ∆ABC ≅ ∆CAD

26. In the figure, ∠E > ∠A and ∠C > ∠D. Then which of the following is a correct relation ?

(a) AD > CE (b) AD < CE (c) AD = CE (d) none of these

27. The measures of the angles of an isosceles triangle in which each of the base angles is four times the vertical angle are :

(a) 80°, 80°, 20° (b) 120°, 30°, 30° (c) 70°, 70°, 40° (d) none of these

28. In the given triangle, AC = AD, ∠CAD = 50° and ∠BAD = ∠23°. Which of the following is true?

(a) AB = BC (b) AB < BC (c) AB > BC (d) none of these

29. A, B, C are three angles of a triangle. If A – B = 25° and B – C = 10°, then ∠ B is :

(a) 60° (b) 45° (c) 55° (d) 40°

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30. In a ∆ABC, the measure of each base angle is 55°. If AB = 5 cm, then the length of side AC is equal to :

(a) 10 cm (b) 8 cm (c) 5 cm (d) 2.5 cm 31. In an isosceles triangle, if the vertical angle is

twice the sum of the base angles, then the vertical angle is :

(a) 130° (b) 100° (c) 120° (d) 110° 32. In the following figure, AD bisects ∠A. Then the

relation between the sides AB, BD and DC is :

(a) AB < DC < AC (b) BD < AB < DC (c) AB > DC > BD (d) DC < BD < AB

33. In ∆ABC, ∠B = 45°, ∠C = 65° and the bisector of ∠BAC meets BC at P. The relation between the sides AP, BP and AB is :

(a) AB > AP > BP (b) AB > BP > AP (c) AB > BP > AP (d) none of these

34. It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true ?

(a) DF = 5 cm, ∠F = 60° (b) DF = 5 cm, ∠E = 60° (c) DE = 5 cm, ∠E = 60° (d) DE = 5 cm, ∠D = 40°

35. Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle cannot be :

(a) 3.6 cm (b) 4.1 cm (c) 3.8 cm (d) 3.4 cm 36. In triangles 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 not isosceles (d) neither congruent nor isosceles

B. Questions From CBSE Examination Papers

1. In ∆ABC, ∠C = ∠A and BC = 6 cm and AC = 5 cm. Then the length of AB is : [T-I (2010)]

(a) 6 cm (b) 5 cm (c) 3 cm (d) 2.5 cm

2. In ∆PQR, ∠P = 60°, ∠Q = 50°. Which side of the triangle is the longest ? [T-I (2010)]

(a) PQ (b) QR (c) PR (d) none 3. In ∆ABC and ∆PQR, AB = PR and ∠A = ∠P.

The two triangles will be congruent by SAS axiom if : [T-I (2010)]

(a) BC = QR (b) AC = PQ (c) AC = QR (d) BC = PR

4. In ∆ABC, ∠A = 50°, ∠B = 60°, arranging the sides of the triangle in ascending order, we get :

[T-I (2010)] (a) AB < BC < CA (b) CA < AB < BC (c) BC < CA < AB (d) BC < AB < CA

5. ∆ABC ≅ ∆PQR. If AB = 5 cm, ∠B = 40° and ∠A = 80°, then which of the following is true ?

[T-I (2010)] (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 : [T-I (2010)]

(a) QE = ER (b) QP > QE (c) QE > QP (d) ER > RP

7. In ∆ABC if AB = BC, then : [T-I (2010)] (a) ∠B > ∠C (b) ∠A = ∠C (c) ∠A = ∠B (d) ∠A < RP 8. If ∆ABC ≅ ∆DEF by SSS congruence rule

then : [T-I (2010)] (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 ? [T-I (2010)]

R

P

Q

125°

100°

(a) PQ = QR (b) PQ > QR (c) PQ < QR (d) ∠P < ∠Q

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10. In triangles ABC and DEF, AB = DE, BC = EF and ∠A = ∠D. Are the triangles congruent ? If yes, by which congruency rule ? [T-I (2010)]

(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

≅ ∆BOC by congurence rule : [T-I (2010)]C B

A D

O

(a) SSS (b) ASA (c) SAS (d) RHS 12. In right triangle DEF, if ∠E = 90°, then : [T-I (2010)] (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 : [T-I (2010)] (a) an isosceles triangle (b) an obtuse angled triangle (c) an equilateral triangle (d) a right triangle 14. Two equilateral triangles are congurent when : [T-I (2010)] (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? [T-I (2010)] (a) QP = 5 cm, ∠P = 60° (b) QP = 5 cm, ∠R = 60° (c) QR = 5 cm, ∠R = 60° (d) QR = 5 cm, ∠Q = 40° 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 :

[T-I (2010)] (a) 80° (b) 75° (c) 100° (d) 135°

17. In the figure, if AB = AC and AP = AQ, then by which congruence criterion ∆PBC ≅ ∆QCB?

[T-I (2010)]

(a) SSS (b) ASA (c) SAS (d) RHS 18. In the figure, in ∆ABC, AB = AC. The value of

x is : [T-I (2010)]A

B Cx

80°

(a) 80° (b) 100° (c) 130° (d) 120° 19. Given ∆OAP ≅ ∆OBP in the figure. The criteria

by which the triangles are congruent is : [T-I (2010)]

O

B

A

P

(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 : [T-I (2010)]

A

B

C

D42°108°

(a) 150° (b) 30° (c) 105° (d) 72° 21. In ∆AOC and ∆XYZ, ∠A = ∠X, AO = XZ, AC

= XY, then by which congruence rule ∆AOC ≅ ∆XZY? [T-I (2010)]

(a) SAS (b) ASA (c) SSS (d) RHS

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22. In the figure, which of the following statements is true ? [T-I (2010)]

A

B C

16 cm 15 cm

19 cm

(a) ∠B = ∠C (b) ∠B is the greatest angle in triangle (c) ∠B is the smallest angle in triangle (d) ∠A is the smallest angle in triangle 23. It is not possible to construct a triangle when its

sides are : [T-I (2010)] (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

24. If ∆ABC ≅ ∆PQR, then which of the following is true ? [T-I (2010)]

(a) AB = RP (b) CA = RP

(c) AC = RQ (d) CB = QP 25. In the given figure, AD is the median, then ∠BAD

is : [T-I (2010)]A

B CD40°

(a) 55° (b) 50° (c) 100° (d) 40°

1. If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles must be congruent. Is the statement true ? Why ?

2. If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent. Is the statement true ? Why?

3. In the two triangles ABC and DEF, AB = DE, and AC = EF. Name two angles from the triangles that must be equal so that the two triangles are congruent. Give reason for your answer.

4. Triangles ABC and DEF are such that AC = 3 cm, BC = 6.5 cm, ∠C = 80°, DE = 3 cm, DF = 6.5 cm and ∠D = 80°. Check whether the two triangles are congruent or not.

5. In ∆ABC and ∆DEF, ∠A = ∠D, ∠B = ∠E and AB = EF. Will the two triangles be congruent ? Justify your answer.

6. Angles X, Y and Z of a triangle are equal. Prove that ∆ XYZ is equilateral.

7. Prove that the sum of three altitudes of a triangle is less than the sum of the three sides of the triangle.

8. AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2 AD. Give reason for your answer.

9. M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that AB + BC + CA > 2 AM ? Give reason for your answer.

10. Arrange the sides of ∆ ABC in ascending order of lengths.

11. Is it possible to construct a triangle with lengths of its sides as 8 cm, 9 cm, and 2 cm,? Give reason for your answer.

12. In an equilateral triangle ABC, if AD is a median, then prove that ∠ADC = 90°.

short Answer type Questions [2 Marks]

a. important Questions

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10. PS is an altitude of an isosceles triangle PQR in which PQ = PR. Show that PS bisects ∠P.

[T-I (2010)] 11. In a ∆DEF, if ∠D = 30°, ∠E = 60°, then which

side of the triangle is longest and which side is shortest ? [T-I (2010)]

12. In the figure, ∠ABD = ∠ACE and AB = AC. Prove that ∆ABD ≅ ∆ACE. [T-I (2010)]

13. In ∆ABC, AB = AC. D is a point inside ∆ABC such that BD = DC. Prove that ∠ABD = ∠ACD.

[T-I (2010)]A

B C

D

14. 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.

[T-I (2010)]A

B C

X Y

15. 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.

[T-I (2010)]A

B C

E F

D

16. In the figure, ABCD is a square and P is the midpoint of AD. BP and CP are joined. Prove that ∠PCB = ∠PBC. [T-I (2010)]

B. Questions From CBSE Examination Papers 1. In the figure, PR = QR, ∠PRA = ∠QRB and ∠BPR

= ∠AQR. Prove that BP = QA. [T-I (2010)]A

B

P R Q

2. In the figure, ∆ABD and ∆BCD are isosceles triangle on the same base BD. Prove that ∠ABC = ∠ADC. [T-I (2010)]

A

B D

C 3. In the figure, ACBD is a quadrilateral with AC

= AD and AB bisects ∠A. Show that ∆ABC ≅ ∆ABD. What can you say about BC and BD ?

[T-I (2010)]

A B

D

C

4. In a ∆ABC, if AB = AC, ∠A = 100°, then find ∠B and ∠C. [T-I (2010)]

5. Prove that each angle of an equilateral angle is 60°. [T-I (2010)]

6. In ∆ABC, AD is perpendicular bisector of BC. Show that ∆ABC is an isosceles triangle in which AB = AC. [T-I (2010)]

7. Prove that in an isosceles triangle, angles opposite to equal sides are equal. [T-I (2010)]

8. In ∆ABC, ∠A = 60°, ∠B = 40°. Which side of this triangle is the smallest? Give reasons for your answer. [T-I (2010)]

9. In the figure, AX = BY and AX||BY, prove that ∆APX ≅ ∆BPY. [T-I (2010)]

X B

P

A Y

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GOYA

L

A B

CD

P

17. In the figure, the diagonal AC of quadrilateral ABCD bisects ∠BAD and ∠BCD. Prove that BC = CD. [T-I (2010)]

A

B

C

D

18. In the figure, ∠B < ∠A and ∠C < ∠D. Show that AD < BC. [T-I (2010)]

B

A

D

C

O

19. Prove that an equilateral triangle can be constructed on any given line segment. [T-I (2010)]

20. In the figure, AB > AC, BO and CO are the bisectors of ∠B and ∠C respectively. Show that OB > OC. [T-I (2010)]

short Answer type Questions [3 Marks]

a. important Questions

1. In the figure, PQ = PR and ∠Q = ∠R. Prove that ∆PQS ≅ ∆PRT.

2. In the figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB = AD and CB = CD.

3. In the figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that ∆ABC ≅ ∆DEF.

4. In the figure, D is any point on the side BC of a triangle ABC. Prove that AB + BC + CA > 2AD.

5. In the figure, AB = AC and ∠1 = ∠2. Prove that ∠PBC = ∠PCB.

6. In the figure, two lines AB and CD intersect each other at O such that BC || DA and BC = DA. Show that O is the mid-point of both the line segments AB and CD.

A

B C

O

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7. In the figure, AD is the bisector of ∠BAC. Prove that AB > BD.

8. D is any point on side AC of a ∆ABC with AB = AC. Show that CD < BD.

9. Four sides of a quadrilateral are equal. Prove that its angles are bisected by its diagonals.

10. In the figure, l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD having its end points on l and m respectively.

B. Questions From CBSE Examination Papers

1. Two sides AB and BC and median AM of ∆ABC are respectively equal to the sides PQ and QR and median PN of ∆PQR. Show that ∆ABM ≅ ∆PQN. [T-I (2010)]

2. In the figure, D is any point on the base BC produced of an isosceles triangle ∆ABC. Prove that AD > AB. [T-I (2010)]

A

B C D

3. In an isosceles triangle ABC with AB = AC, BD and CE are two medians. Prove that BD = CE.

[T-I (2010)] 4. In the figure, if PS = PR, ∠TPS = ∠QPR, then

prove that PT = PQ. [T-I (2010)]T S R Q

P

5. In ∆ABC, BD and CE are two altitudes such that BD = CE. Prove that ∆ABC is isosceles.

[T-I (2010)]

6. 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. [T-I (2010)] 7. D is a point on side BC of ∆ABC such that AD

= AC. Show that AB > AD. [T-I (2010)] 8. In right triangle ABC, ∠C = 90°, M is midpoint 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 : [T-I (2010)]

(i) ∆AMC ≅ ∆BMD (ii) ∠DBC = ∠ACB 9. In the figure, PR > PQ and PS bisects ∠PQR.

Prove that ∠PSR > ∠PSQ. [T-I (2010)]

SQ

P

R

10. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that AD is also the median of the triangle. [T-I (2010)]

11. In the figure, ∆ABC and ∆DBC are two isosceles triangles on the same base BC and the vertices A and D are on the same side of BC. AD is extended to meet BC at P. Prove that AP bisects ∠A.

[T-I (2010)]

PB

A

C

D

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12. 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. [T-I (2010)]

E

B

A

CD

F

13. Prove that medians of an equilateral triangle are equal. [T-I (2010)]

14. In the figure, line segment AB||CD and O is the mid point of AD. Show that : [T-I (2010)]

(i) ∆AOB ≅ ∆DOC. (ii) O is also the mid point of BC.

C D

A B

O

15. In the figure, AD and BC are equal and perpendiculars to the same line segment AB. Show that CD bisects AB. [T-I (2010)]

B C

O

D A

16. In the figure, BD and CE are two altitudes of ∆ABC such that BD = CE. Prove that ∆ABC is isosceles. [T-I (2010)]

D

B

A

C

E

17. Show that in a right angled triangle the hypotenuse is the longest side. [T-I (2010)]

18. In the figure, if ∠a > ∠b, then prove that PQ > PR. [T-I (2010)]

19. 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. [T-I (2010)]

A

B C

D

20. 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. [T-I (2010)]

A

B D E C

21. In the figure, ∠B = ∠C and AB = AC. Prove that BE = CF. [T-I (2010)]

22. 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. [T-I (2010)]

A

B C

D

P

23. In the figure, ∆LMN is an isosceles triangle with LM = LN and LP bisects ∠NLQ. Prove that LP||MN. [T-I (2010)]

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long Answer type Questions [4 Marks]

a. important Questions

Q

M

LP

N

24. 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.

[T-I (2010)]R

P Q

O

N M

25. In the figure, ∆ABC is an isosceles triangle in which AB = AC, side BA is produced to D such that AD = AB. Show that ∠BCD is a right angle.

[T-I (2010)]

D

C

A

B 26. In the figure, D is a point on side BC of ∆ABC

such that AD = AC. Show that AB > AD. [T-I (2010)]

A

B CD 27. 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.

[T-I (2010)]

1. Prove that if in two triangles two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the triangles are congruent.

2. O is any point in the interior of ∆ABC. Show that OB + OC < AB + AC

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

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

5. ABC is a right angled triangle with AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.

6. Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2 (BD + AC).

7. In a triangle ABC, D is the mid-point of side AC

such that BD = 12

AC. Show that ∠ABC is a right

angle.

8. ABC is a triangle in which ∠B = 2∠C. D is a point on side BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.

9. ABCD is a quadrilateral such that AB = AD and CB = CD. Prove that AC is the perpendicular bisector of BD.

B. Questions From CBSE Examination Papers

1. In the figure, A is a point equidistant from two lines l1 and l2 intersecting at a point P. Show that AP bisects the angle between l1 and l2. [T-I (2010)]

l1

P

A

l2

2. In a triangle PQR, PR > PQ and PS is the bisector of ∠QPR. Prove that ∠PSR > ∠PSQ.

[T-I (2010)] 3. In the figure, two sides AB and BC and the median

AM of ∆ABC are respectively equal to sides DE and EF and the median DN of ∆DEF. Prove that ∆ABC ≅ ∆DEF. [T-I (2010)]

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A

B M C

D

E N F

4. In the figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE. [T-I (2010)]

5. 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.

[T-I (2010)]P

A B

Q

C

6. In ∆’s ABC and PQR, AB = PQ, AC = PR and altitude AM and altitude PN are equal. Show that ∆ABC ≅ ∆PQR. [T-I (2010)]

7. Prove that the sum of any two sides of a triangle is greater than twice the length of median drawn to the third side. [T-I (2010)]

8. In the figure, show that 2(AC + BD) > (AB + BC + CD + DA) [T-I (2010)]

D C

O

A B

9. In the figure, if ∠x = ∠y and AB = AC, then prove that AD = AE. [T-I (2010)]

BD

EC

Ax

yO

10. O is a point in the interior of ∆PQR. Prove that

OP + OQ + OR > 12

(PQ + QR + PR). [T-I (2010)] 11. 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 : [T-I (2010)]

(i) OB = OC (ii) AO bisects ∠A 12. Show that perimeter of a triangle is greater than

the sum of its medians. [T-I (2010)] 13. In the figure, ∠BCD = ∠ADC and ∠ACB = ∠BDA.

Prove that AD = BC and ∠A = ∠B. [T-I (2010)]A B

C D

14. 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.

[T-I (2010)]

A

D

B C

15. 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. [T-I (2010)]

A

B C

D

P

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16. 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. [T-I (2010)]

A

B D

M

C

L

17. Prove that two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle. [T-I (2010)]

18. 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. [T-I (2010)]

19. In the figure, AC = BC, ∠DCA = ∠ECB and ∠DBC = ∠EAC. Prove that (i) ∆DBC ≅ ∆EAC; (ii) DC = EC and BD = AE. [T-I (2010)]

D E

C BA

20. In ∆ABC, AB = AC, and the bisectors of ∠B and ∠C intersect at point O. Prove that BO = CO and the ray AO is the bisector of ∠BAC.

[T-I (2010)] 21. In the figure, BA||PQ, CA||RS and BP = RC. Prove

that (i) BS = PQ; (ii) RS = CQ. [T-I (2010)]A

S Q

B P R C

22. 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 : [T-I (2010)]

(i) ∆AMC ≅ ∆BMD (ii) ∠DBC = 90° (iii) ∆DBC ≅ ∆ACB

M

D A

B C

23. 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.

[T-I (2010)]R

P Q

O

N M

24. In the figure, ABCD is a square and ∆DEC is an equilateral triangle. Prove that :

(i) ∆ADE ≅ ∆BCE, (ii) AE = BE, (iii) ∠DAE = 15°. [T-I (2010)]

E

D C

A B 25. In the figure, S is any point in the interior of ∆PQR.

Show that SQ + SR < PQ + PR. [T-I (2010)]P

Q

S

R

26. In the figure, PQ and RS are perpendicular to QS, QA = BS and PB = AR. Prove that ∠QPB = ∠SRA. [T-I (2010)]

P R

Q A B S

27. In figure, AP and DP are bisectors of two adjacent angles A and D of a quadrilateral ABCD. Prove that 2∠APD = ∠B + ∠C. [T-I (2010)]

AB

CD

P

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