M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 10 triangles

It is given that ∆ABC ≅ ∆RPQ. Is it true to say that BC = QR? Why?

In the given figure, AB=AC and AP=AQ. Prove that

(i) ∆APC ≅ ∆AQB

(ii) CP = BQ

(iii) ∠APC = ∠AQB.

In the given figure, AB = AC, P, and Q are points on BA and CA respectively such that

AP = AQ. Prove that

(i) ∆APC ≅ ∆AQB

(ii) CP = BQ

(iii) ∠ACP = ∠ABQ.

In the given figure, AD = BC and BD = AC. Prove that :

∠ADB = ∠BCA and ∠DAB = ∠CBA

In the given figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA.

Prove that

(i) ∆ABD ≅ ∆BAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC.

In the given figure, AB = DC and AB || DC. Prove that AD = BC.

In the given figure. AC = AE, AB = AD and ∠BAD = ∠CAE. Show that BC = DE.

In the adjoining figure, AB = CD, CE = BF and ∠ACE = ∠DBF. Prove that

(i) ∆ACE ≅ ∆DBF

(ii) AE = DF.

In triangles ABC and PQR, ∠A= ∠Q and ∠B = ∠R. Which side of APQR should be

equal to side AB of AABC so that the two triangles are congruent? Give the reason for your

answer.

In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of APQR should be

equal to side BC of AABC so that the two triangles are congruent? Give the reason for your

answer.

“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?

In the given figure, AD is median of ∆ABC, BM and CN are perpendiculars drawn

from B and C respectively on AD and AD produced. Prove that BM = CN.

In the given figure, BM and DN are perpendiculars to the line segment AC. If BM =

DN, prove that AC bisects BD.

In the given figure, l and m are two parallel lines intersected by another pair of

parallel lines p and q. Show that ∆ABC ≅ ∆CDA.

In the given figure, two lines AB and CD intersect each other at the point O such

that BC || DA and BC = DA. Show that O is the mid-point of both the line segments AB

and CD.

ABC is a right-angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

Show that every equiangular triangle is equilateral.

In the following diagrams, find the value of x:

(a) In the figure (1) given below, ABC is an equilateral triangle. Base BC is produced

to E, such that BC’= CE. Calculate ∠ACE and ∠AEC.

(b) In the figure (2) given below, prove that ∠ BAD: ∠ ADB = 3: 1.

(c) In the figure (3) given below, AB || CD. Find the values of x, y, and ∠.

In the given figure, AD, BE, and CF arc altitudes of ∆ABC. If AD = BE = CF, prove that

ABC is an equilateral triangle.

In the given figure, D is the mid-point of BC, DE and DF are perpendiculars to AB and AC

respectively such that DE = DF. Prove that ABC is an isosceles triangle.

In ∆PQR, ∠P = 70° and ∠R = 30°. Which side of this triangle is the longest? Give reason

for your answer.

Show that in a right-angled triangle, the hypotenuse is the longest side.

PQR is a right angle triangle at Q and PQ: QR = 3:2. Which is the least angle?

In ∆ ABC, AB = 8 cm, BC = 5.6 cm and CA = 6.5 cm. Which is

(i) the greatest angle ?

(ii) the smallest angle?