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**FILL IN THE BLANKS :**

(a) Sides opposite to the equal angles of a triangle are______.

(b) Angles opposite to the equal sides of a triangle are______.

(c) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is ______CE.

(d) If altitudes CE and BF of a triangle ABC are equal, then AB =______.

(e) In right triangles PQR and DEF, if hypotenuse PQ = hypotenuse EF and side PR = DE, then ΔPQR Δ______.

(f) In a triangle ABC, if BC = AB and ∠C = 80°, then ∠B =______.

(g) In a triangle PQR, if ∠P = ∠R, then PQ =______.

(h) If two sides and the______angle of one triangle are respectively equal to two sides and the included angle

of the other triangle, then the triangles are congruent.

(i) If______sides of a triangle are respectively equal to the three sides of the other triangle, then the triangles are congruent.

(j) If in two triangles ABC and PQR, AB = QR, ∠A = ∠Q and ∠B = ∠R, then ΔABC Δ______.

(k) If in two triangles ABC and DEF, AB = DF, BC = DE and ∠B = ∠D, then ΔABC Δ______.

(l) If in two triangles PQR and DEF, PR = EF, QR = DE and PQ = FD, then ΔPQR Δ______.

(m) Sum of any two sides of a triangle is______than the third side.

(n) If two angles of a triangle are unequal, then the smaller angle has the______side opposite to it.

(o) Of all the line segments drawn from a point to a line not containing it, the______line segment is the shortest.

(p) Difference of any two sides of a triangle is______than the third side.

(q) If any two sides of a triangle are unequal, then the larger side has the______angle opposite to it.

(r) The sum of the three altitudes of a triangle is______than its perimeter.

(s) In a right triangle, the hypotenuse is the______side.

(t) The perimeter of a triangle is______than the sum of its medians.

**VERY SHORT ANSWER TYPE QUESTIONS :**

1. Which of the following pairs of triangles are congruent?

(a) ∠ABC and ∠DEF in which : BC = EF, AC = DF and ∠C = ∠F.

(b) ∠ABC and ∠PQR in which : AB = PQ, BC = QR and ∠C = ∠R.

(c) ∠ABC and ∠LMN in which : ∠A = ∠L = 90°, AB = LM , ∠C = 40° and ∠M = 50°.

(d) ∠ABC and ∠DEF in which : ∠B = ∠E = 90° and AC = DF.

2. Answer the following as per the exact requirement :

(a) In Δs ABC and PQR, AB = PQ, AC = PR

and ∠BAC = ∠QPR.

Here, ΔABC ΔPQR.

Justify the statement by writing the congruence R

criteria applicable in this situation.

(b) In fig. ∠BAC = ∠QRP.

Justify that ΔABC ΔRQP.

3. In ΔABC, AB = AC. OB and OC are bisectors of ∠B and ∠C respectively. Show that OB = OC.

4. In fig, ∠1 > ∠2. Show that AB > AC.

5. In ΔABC, we have, ∠A > ∠B > ∠C, then determine the shortest and the longest side of the triangle.

6. If ΔABC ΔPQR, ∠B = 40° and ∠C = 95°, find ∠P.

7. In ΔABC, AB = BC = 5cm and ∠A = 55°, find ∠B.

8. State the angle-angle-side congruence criteria for triangles.

9. In fig, AB = AC and ∠ACD = 115°. Find ∠A.

10. In ΔABC, BC = AC and ∠B = 64°, find ∠C.

11. In ΔPQR, ∠P = 50° and ∠R = 70°. Name : (i) the shortest side (ii) the longest side of the triangle.

** **

**SHORT ANSWER TYPE QUESTIONS :**

1. In the given fig, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.

2. In the given fig. AY ⊥ ZY and BY ⊥ XY such that AY = ZY and BY = XY. Prove that AB = ZX.

3. If the bisector of the exterior vertical angle of a triangle is parallel to the base, show that the triangle is isosceles.

4. In each of the following figures, find the value of x:

(i) (ii)

5. In each of the following figures, find the value of x:

(i) (ii)

6. In the given fig, BDCE; AC = BC, ∠ABD = 20° and ∠ECF = 70°. Find ∠GAC.

7. In the given figure, ABCD and CA = CE. Find the values of x, y and z.

8. In the given figure, AB = AD; CB = CD; ∠A = 42° and ∠C = 108°, find ∠ABC.

9. In the given figure, side BA of ΔABC has been produced to D such that CD = CA and side CB has been produced to E. If ∠BAC = 106° and ∠ABE = 128°, find ∠BCD.

10. In the given figure, AB = BC and AC = CD. Show that ∠BAD : ∠ADB = 3 : 1.

11. In the given figure, AD is the internal bisector of ∠A and CEDA. If CE meets BA produced at E, prove that ΔCAE is isosceles.

12. In the given figure, AD bisects ∠A. Arrange AB, BD and DC in ascending order.

13. In the given fig. AB = AC. Prove that : AF > AE.

14. In the given figure, side AB of ΔABC is produced to D such that BD = BC.

If ∠A = 60° and ∠B = 50°, prove that :

(i) AD > CD

(ii) AD > AC

15. In the given figure, AD bisects ∠A. If ∠B = 60°, ∠C = 40°, then arrange AB, BD and DC in ascending order of their lengths.

(B) LONG ANSWER TYPE QUESTIONS :

1. In the given fig, ABCD is a square and ΔPAB is an equilateral triangle.

(i) Prove that ΔAPD ΔBPC.

(ii) Show that ∠DPC = 15°.

2. In the given fig, in ΔABC, ∠B = 90°. if ABPQ and ACRS are squares, prove that :

(i) ΔACQ ΔABS.

(ii) CQ = BS.

3. Squares ABPQ and ADRS are drawn on the sides AB and AD of a parallelogram ABCD. Prove that :

(i) ∠SAQ = ∠ABC

(ii) SQ = AC.

4. In the given fig, ABCD is a square and P, Q, R are points on AB, BC and CD respectively such that AP = BQ= CR and ∠PQR = 90°. Prove that : (i) PB = QC, (ii) PQ = QR, (iii) ∠QPR = 45°.

5. In the given fig, ABCD is a square, EFBD and R is the mid-point of EF. Prove that :

(i) BE = DF

(ii) AR bisects ∠BAD

(iii) If AR is produced, it will pass through C.

6. In a ΔABC, AB = AC and BC is produced to D. From D, DE is drawn perpendicular to BA produced and DF is drawn perpendicular to AC produced. Prove that BD bisects ∠EDF.

7. Prove that the perimeter of a triangle is greater than the sum of its three medians.

8. In the adjoining figure, prove that :

(i) AB + BC + CD > DA

(ii) AB + BC + CD + DA > 2AC

(iii) AB + BC + CD + DA > 2BD

(iv) AB + BC + CD + DA > AC + BD

9. In the adjoining figure, O is the centre of a circle, XY is a diameter and XZ is a chord. Prove that XY > XZ.

10. In the given figure, AD = AB and AE bisects ∠A. Prove that :

(i) BE = ED

(ii) ∠ABD > ∠BCA.

**ANSWER KEY**

VERY SHORT ANSWER TYPE QUESTIONS :

1. (a), (c)

2. (a) SAS congruence criteria

5. Shortest side is AB and the longest side is BC.

6. 45°

7. 70°

9. 50°

10. 52°

11. (i) QR, (ii) PQ

**SHORT ANSWER TYPE QUESTIONS :**

4. (i) 110, (ii) 55

5. (i) 22, (ii) 40

6. 130°

7. x = 36, y = 68, z = 44

8. 105°

9. 54°

12. BD < AB < DC

15. BD = DC < AB

**FILL IN THE BLANKS :**

(a) Equal

(b) Equal

(c) Equal to

(d) AC

(e) EFD

(f) 20°

(g) RQ

(h) Included

(i) Three

(j) QRP

(k) FDE

(l) FDE

(m) Greater

(n) Smaller

(o) Perpendicular

(p) Less

(q) Greater

(r) Less

(s) Largest

(t) Greater