Selina solutions concise maths class 9 icse Solutions for Chapter 9 chapter 9 triangles congruency in triangles

Which of the following pairs of triangles are congruent? In each case, state the

condition of congruency:

(a) In ? ABC and ? DEF, AB=DE, BC=EF and ∠B=∠E.

(b) In ? ABC and ? DEF, ∠B=∠E = 90° ; AC=DF and BC=EF.

(c) In ? ABC and ? QRP, AB=QR, ∠B=∠R and ∠C=∠P.

(d) In ? ABC and ? PQR, AB=PQ, AC=PR, and BC=QR.

(e) In ? ABC and ? PQR, BC=QR, ∠A=90°, ∠C=∠R=40o and ∠Q=50°.

The given figure shows a circle with center O. P is mid-point of chord AB.

The following figure shows a circle with center O.

In a triangle ABC, D is mid-point of BC; AD is produced up to E so that DE =

AD. Prove that:

(i) ? ABD and ? ECD are congruent.

(ii) AB=EC

(iii) AB is parallel to EC.

A triangle ABC has ∠B=∠C.

Prove that:

(i) The perpendiculars from the mid-point of BC to AB and AC are equal.

(ii) The perpendiculars form B and C to the opposite sides are equal.

The perpendicular bisector of the sides of a triangle AB meet at I.

Prove that: IA=IB=IC.

A line segment AB is bisected at point P and through point P another line

segment PQ, which is perpendicular to AB, is drawn. Show that: QA=QB.

If AP bisects angle BAC and M is any point on AP, prove that the

perpendiculars drawn from M to Ab and AC are equal.

From the given diagram, in which ABCD is a parallelogram, ABL is a line

segment and E is mid-point of BC.

Prove that:

In the given figure, AB=DB and AC=DC.

In the given figure: AB/FD,AC//GE and BD=CE; prove that:

(i) BG=DF

(ii) CF=EG

In a triangle ABC, AB=AC. Show that the altitude AD is median also.

In the following figure, AB=AC and AD is perpendicular to BC. BE bisects angle

B and EF is perpendicular to AB.

Prove that:

(i) BD=CD

(ii) ED=EF

Use the information in the given figure to prove:

(i) AB=FE

(ii) BD=CF

On the sides AB and AC of triangle ABC, equilateral triangle ABD and ACE are

drawn. Prove that:

In the following diagrams, ABCD is a square and APB is an equilateral triangle.

In each case,

In the figure, given below, triangle ABC is a right-angled triangle at B. ABPQ and

ACRS are squares. Prove that:

(i) ?ACQ and ?ASB are congruent.

(ii) CQ=BS

In a ? ABC, BD is the median to the side AC, BD is produced to E such that

BD=DE.

Prove that: AE is parallel to BC.

In the adjoining figure, QX and RX are the bisectors of the angles Q and R

respectively of the triangle PQR.

If XS⊥QR and XT⊥PQ; prove that:

In parallelogram ABCD, the angles A and C are obtuse. Points X and Y are

taken on the diagonal BD such that the angles XAD and YCB are right angles.

Prove that: XA=YC.

ABCD is a parallelogram. The sides AB and AD are produced to E and F

respectively, such that AB=BE and AD=DF.

Prove that: ?BEC≅ ?DCF

In the following figures, the sides AB and BC and the median AD of triangle

ABC are respectively equal to the sides PQ and QR and the median PS of the

triangle PQR. Prove that ?ABC and ?PQR are congruent.

In the following figure, OA=OC and AB=BC.

Prove that:

In the following diagram, AP and BQ are equal and parallel to each other.

The following figure shows a triangle ABC in which AB=AC. M is a point on AB

and N is a point on AC such that BM=CN.

Prove that:

(i) AM=AN

(ii) ?AMC ≅ ?ANB

(iii) BN=CM

(iv) ?BMC ≅ ?CNB

In a triangle, ABC, AB=BC, AD is perpendicular to side BC and CE is

perpendicular to side AB. Prove that:

AD=CE.

PQRS is a parallelogram. L and M are points on PQ and SR respectively such

that PL=MR. Show that LM and QS bisect each other.

In the following figure, ABC is an equilateral triangle in which P is parallel to

AC. Side AC is produced up to point R so that CR=BP. Prove that QR bisects

PC.

In the following figure ∠A=∠C and AB=BC.

AD and BC are equal perpendiculars to a line segment AB. If AD and BC are

on different sides of AB prove that CD bisects AB.

In ?ABC, AB=AC and the bisectors of angles B and C intersect at point O.

Prove that:

(i) BO=CO

(ii) AO bisects angle BAC.

A point O is taken inside a rhombus ABCD such that its distance from the

vertices B and D are equal. Show that AOC is a straight line.

In quadrilateral ABCD, AD=BC and BD=CA. Prove that:

(i) ∠ADB = ∠BCA

(ii) ∠DAB = ∠CBA