# Triangles

**Exercise 1**

**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

**Exercise 2**

**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 = 90^{0} and AB = AC. Find ∠ B and ∠ C.

**16. ** Show that the angles of an equilateral triangle are 60^{0} each.

**Exercise 3**

**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.

**Exercise 4**

**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.

**Exercise 5**

**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 A

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?