1. In quadrilateral ABCD (See figure). AC = AD and AB bisects ∠ A. Show that Δ ABC ≅ Δ ABD. What can you say about BC and BD?
2. ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA. (See figure). Prove that:-
(i) Δ ABD ≅ Δ BAC
(ii) BD = AC
(iii) ∠ ABD = ∠ BAC
3. AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB (See figure)
4. l and m are two parallel lines intersected by another pair of parallel lines p and q (See figure). Show that Δ ABC ≅ Δ CDA.
5. Line l is the bisector of the angle A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A. Show that:
(i) Δ APB ≅ Δ AQB
(ii) BP = BQ or P is equidistant from the arms of ∠ A (See figure).
6. In figure, AC = AB, AB = AD and ∠ BAD = ∠ EAC. Show that BC = DE.
7. AB is a line segment and P is the mid-point. D and E are points on the same side of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB. Show that:
(i) Δ DAF ≅ Δ FBP
(ii) AD = BE (See figure)
8. In 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. (See figure) Show that:
(i) Δ AMC ≅ Δ BMD
(ii) ∠ DBC is a right angle.
(ii) Δ DBC ≅ Δ ACB
9. In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that:
(i) OB = OC
(ii) AO bisects ∠ A.
10. In Δ ABC, AD is the perpendicular bisector of BC (See figure). Show that Δ ABC is an isosceles triangle in which AB = AC.
11. ABC is an isosceles triangle in which altitudes BE and CF are drawn to sides AC and AB respectively (See figure). Show that these altitudes are equal.
12. ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (See figure). Show that:
(i) Δ ABE ≅ Δ ACE
(ii) AB = AC or A ABC is an isosceles triangle.
13. ABC and DBC are two isosceles triangles on the same base BC (See figure). Show that ∠ ABD = ∠ ACD.
14. Δ 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 (See figure).
15. ABC is a right angled triangle in which ∠ A = 900 and AB = AC. Find ∠ B and ∠ C.
16. Show that the angles of an equilateral triangle are 600 each.
17. Δ ABC and Δ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (See figure). If AD is extended to intersect BC at P, show that:
(i) Δ ABD ≅ Δ ACD
(ii) Δ ABP ≅ Δ ACP
(iii) AP bisects ∠ A as well as ∠ D.
(iv) AP is the perpendicular bisector of BC.
18. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that:
(i) AD bisects BC.
(ii) AD bisects ∠ A.
19. Two sides AB and BC and median AM of the triangle ABC are respectively equal to side PQ and QR and median PN of Δ PQR (See figure). Show that:
(i) Δ ABM ≅ Δ PQN
(ii) Δ ABC ≅ Δ PQR
20. BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.
21. ABC is an isosceles triangles with AB = AC. Draw AP ⊥ BC and show that ∠ B = ∠ C.
22. Show that in a right angles triangle, the hypotenuse is the longest side.
23. In figure, sides AB and AC of Δ ABC are extended to points P and Q respectively. Also ∠ PBC < ∠ QCB. Show that AC > AB.
24. In figure, ∠B < ∠A and ∠C < ∠D. Show that AD < BC. B
25. AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (See figure). Show that ∠A > ∠C and ∠B > ∠D.
26. In figure, PR > PQ and PS bisects ∠QPR. Prove that ∠PSR > ∠PSQ.
27. Show that all the line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
28. ABC is a triangle. Locate a point in the interior of Δ ABC which is equidistant from all the vertices of Δ ABC.
29. In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.
30. In a huge park, people are concentrated at three points (See figure).
A: where there are different slides and swings for children.
B: near which a man-made lake is situated.
C: which is near to a large parking and exit.
Where should an ice cream parlour be set up so that maximum number of persons can approach it?
Û B Û C
31. Complete the hexagonal rangoli and the star rangolies (See figure) bu filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?