1. PQ = PR of < QPR and S and T are point on PR and PQ such that ∠PQS = ∠PRT . Prove that Δ PQS ≅ Δ PRT.

2. Two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the midpoint of both the line-segments AB and CD( join B-C and A-D)

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

4. ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE.

5. D and E are points on side BC of a Δ ABC such that BD = CE and AD = AE. Show that Δ ABD ≅ Δ ACE.

6. CDE is an equilateral triangle formed on a side CD of a square ABCD (join AE and BE). Show that Δ ADE ≅ Δ BCE.

7. BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that Δ ABC ≅ Δ DEF

.8. Q is a point on the side SR of a Δ PSR such that PQ = PR. Prove that PS > PQ.

9. S is any point on side QR of a Δ PQR. Show that: PQ + QR + RP > 2 PS

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

11. 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.

12. Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Prove that ∠MOC =∠ABC.

13. Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.

14. AD is the bisector of ∠BAC. of DABC . Prove that AB > BD.

15. ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. AD perpendicular to BC. Show that hypotenuse AC = 2 BC.

16. 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 two triangles are congruent.

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

18. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR. { Produce QS to intersect PR at T}

**CBSE TEST PAPER-2**

**1 marks questions**

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

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

2 . If AB = QR, BC = PR and CA = PQ, then

(A) Δ ABC ≅ Δ PQR (B) Δ CBA ≅ Δ PRQ (C) Δ BAC ≅ Δ RPQ (D) Δ PQR ≅ Δ BCA

3 . In Δ ABC, AB = AC and ∠B = 50°. Then ∠C is equal to

(A) 40° (B) 50° (C) 80° (D) 130°

**2 marks questions**

1. In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of Δ PQR should be equal to side AB of Δ ABC so that the two triangles are congruent? Give reason for your answer.

2 . In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of Δ PQR should be equal to side BC of Δ ABC so that the two triangles are congruent? Give reason for your answer.

3 . AB is a line segment and line l is its perpendicular bisector. If a point P lies on l, show that P is equidistant from A and B.

**3 marks questions**

1. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR.

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

3. P is a point on the bisector of ∠ABC. If the line through P, parallel to BA meets BC at Q, prove that BPQ is an isosceles triangle.

**4 marks questions**

1. Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side

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

3. In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.