In this page we have *NCERT book Solutions for Class 9th Maths:Triangle* for
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We must revise these notes before proceding with the questionsAAA( Angle Angle Angle) is not the right condition to Prove congurence.

How to Prove the congurence of Two Triangle

1) We have already studied that two triangles are congurent when all the sides and all the angles are equal. But we dont need prove all these while solving the Problem

2) We just need to prove the congurence using the different criterio like SSS,ASA,SAS,RHS,AAS

3) Dont use AAA

4) Use the theorem learn in previous Geometry chapter like vertically opposite angles,alternate interior angles,corresonding angles

5)Write down the corresponding angles and corresponding sides carefully

6) We need to be careful with the labelling when our Triangles are in different positions

**Question 1**

In quadrilateral ACBD, AC = AD and AB bisects ∠A (see below figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD?

**Answer**

__Given__,

AC = AD and AB bisects ∠A

__To prove__,

ΔABC ≅ ΔABD

__Proof__,
In ΔABC and ΔABD,

AB = AB (Common)

AC = AD (Given)

∠CAB = ∠DAB (AB is bisector)

By SAS (Side-Angle-Side) congruence condition.

Therefore, ΔABC ≅ ΔABD.

Now from CPCT, we know that

BC=BD

**Question 2**

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC.

**Answer**

Given,

AD = BC and ∠DAB = ∠CBA

(i) In ΔABD and ΔBAC,

AB = BA (Common)

∠DAB = ∠CBA (Given) AD = BC (Given)

By SAS congruence condition.

So, ΔABD ≅ ΔBAC

(ii) Since, ΔABD ≅ ΔBAC

Therefore BD = AC by CPCT

(iii) Since, ΔABD ≅ ΔBAC

Therefore ∠ABD = ∠BAC by CPCT

**Question 3 **

AD and BC are equal perpendiculars to a line segment AB (see below figure). Show that CD bisects AB.

**Answer**

__Given,__

AD and BC are equal perpendiculars to AB.

__To prove__,

CD bisects AB

__Proof__,

In ΔAOD and ΔBOC,

∠A = ∠B (As Perpendicular)

∠AOD = ∠BOC (Vertically opposite angles)

AD = BC (Given)

By AAS(Angle-Angle-Side) congruence condition.

So, ΔAOD ≅ ΔBOC

Now by CPCT

AO = OB

So CD bisects AB.

**Question 4**

l and m are two parallel lines intersected by another pair of parallel lines p and q (see below figure). Show that ΔABC ≅ ΔCDA.

**Answer**

__Given__,

l || m and p || q

__To prove__,

ΔABC ≅ ΔCDA

__Proof,__

In ΔABC and ΔCDA,

∠BCA = ∠DAC (Alternate interior angles) AC = CA (Common)

∠BAC = ∠DCA (Alternate interior angles)

By ASA(Angle-Side-Angle) congruence condition. So, ΔABC ≅ ΔCDA

**Question 5**

Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see below figure). Show that:

(i) ΔAPB ≅ ΔAQB

(ii) BP = BQ or B is equidistant from the arms of ∠A.

**Answer**

Given,

l is the bisector of an angle ∠A.

BP and BQ are perpendiculars.

(i) In ΔAPB and ΔAQB,

∠P = ∠Q (Right angles)

∠BAP = ∠BAQ (l is bisector)

AB = AB (Common)

Therefore, ΔAPB ≅ ΔAQB by AAS congruence condition.

(ii) BP = BQ by CPCT. Therefore, B is equidistant from the arms of ∠A.

**Question 6**

In below figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.

**Answer**

__Given,__

AC = AE, AB = AD and ∠BAD = ∠EAC

__To Prove__,

BC = DE

Proof,

∠BAD = ∠EAC

By Adding ∠DAC both sides

∠BAD + ∠DAC = ∠EAC + ∠DAC ∠BAC = ∠EAD

In ΔABC and ΔADE,

AC = AE (Given)

∠BAC = ∠EAD AB = AD (Given)

By SAS (Side -Angle-Side) congruence condition.

So, ΔABC ≅ ΔADE

By CPCT

BC = DE

**Question 7**

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠BAD = ∠ABE and ∠EPA = ∠DPB (see below figure). Show that

(i) ΔDAP ≅ ΔEBP

(ii) AD = BE

**Answer**

Given,

P is mid-point of AB.

∠BAD = ∠ABE and ∠EPA = ∠DPB

(i) ∠EPA = ∠DPB

By Adding ∠DPE both sides

∠EPA + ∠DPE = ∠DPB + ∠DPE ∠DPA = ∠EPB

In ΔDAP ≅ ΔEBP,

∠DPA = ∠EPB

AP = BP (P is mid-point of AB)

∠BAD = ∠ABE (Given)

By ASA congruence condition. So, ΔDAP ≅ ΔEBP

(ii) By CPCT

AD = BE

**Question 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 below figure). Show that:

(i) ΔAMC ≅ ΔBMD

(ii) ∠DBC is a right angle.

(iii) ΔDBC ≅ ΔACB

(iv) CM = 1/2 AB

**Answer**

Given,

∠C = 90°, M is the mid-point of AB and DM = CM

(i) In ΔAMC and ΔBMD,

AM = BM (M is the mid-point)

∠CMA = ∠DMB (Vertically opposite angles)

CM = DM (Given)

By SAS(Side-Angle-Side) congruence condition

So, ΔAMC ≅ ΔBMD.

(ii)By CPCT

∠ACM = ∠BDM

Therefore, AC || BD as alternate interior angles are equal.

Now,

∠ACB + ∠DBC = 180° (co-interiors angles)

90° + ∠B = 180°

∠DBC = 90°

(iii) In ΔDBC and ΔACB,

BC = CB (Common)

∠ACB = ∠DBC (Right angles)

DB = AC (by CPCT, already proved)

By (Side-Angle-Side) congruence condition.

So, ΔDBC ≅ ΔACB

(iv) DC = AB (ΔDBC ≅ ΔACB)

DM = CM = AM = BM (M is mid-point)

DM + CM = AM + BM CM + CM = AB

CM = 1/2AB

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Class 9 Maths Class 9 Science

1) We have already studied that two triangles are congurent when all the sides and all the angles are equal. But we dont need prove all these while solving the Problem

2) We just need to prove the congurence using the different criterio like SSS,ASA,SAS,RHS,AAS

3) Dont use AAA

4) Use the theorem learn in previous Geometry chapter like vertically opposite angles,alternate interior angles,corresonding angles

5)Write down the corresponding angles and corresponding sides carefully

6) We need to be careful with the labelling when our Triangles are in different positions

In quadrilateral ACBD, AC = AD and AB bisects ∠A (see below figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD?

AC = AD and AB bisects ∠A

ΔABC ≅ ΔABD

AB = AB (Common)

AC = AD (Given)

∠CAB = ∠DAB (AB is bisector)

By SAS (Side-Angle-Side) congruence condition.

Therefore, ΔABC ≅ ΔABD.

Now from CPCT, we know that

BC=BD

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC.

Given,

AD = BC and ∠DAB = ∠CBA

(i) In ΔABD and ΔBAC,

AB = BA (Common)

∠DAB = ∠CBA (Given) AD = BC (Given)

By SAS congruence condition.

So, ΔABD ≅ ΔBAC

(ii) Since, ΔABD ≅ ΔBAC

Therefore BD = AC by CPCT

(iii) Since, ΔABD ≅ ΔBAC

Therefore ∠ABD = ∠BAC by CPCT

AD and BC are equal perpendiculars to a line segment AB (see below figure). Show that CD bisects AB.

AD and BC are equal perpendiculars to AB.

CD bisects AB

In ΔAOD and ΔBOC,

∠A = ∠B (As Perpendicular)

∠AOD = ∠BOC (Vertically opposite angles)

AD = BC (Given)

By AAS(Angle-Angle-Side) congruence condition.

So, ΔAOD ≅ ΔBOC

Now by CPCT

AO = OB

So CD bisects AB.

l and m are two parallel lines intersected by another pair of parallel lines p and q (see below figure). Show that ΔABC ≅ ΔCDA.

l || m and p || q

ΔABC ≅ ΔCDA

In ΔABC and ΔCDA,

∠BCA = ∠DAC (Alternate interior angles) AC = CA (Common)

∠BAC = ∠DCA (Alternate interior angles)

By ASA(Angle-Side-Angle) congruence condition. So, ΔABC ≅ ΔCDA

Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see below figure). Show that:

(i) ΔAPB ≅ ΔAQB

(ii) BP = BQ or B is equidistant from the arms of ∠A.

Given,

l is the bisector of an angle ∠A.

BP and BQ are perpendiculars.

(i) In ΔAPB and ΔAQB,

∠P = ∠Q (Right angles)

∠BAP = ∠BAQ (l is bisector)

AB = AB (Common)

Therefore, ΔAPB ≅ ΔAQB by AAS congruence condition.

(ii) BP = BQ by CPCT. Therefore, B is equidistant from the arms of ∠A.

In below figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.

AC = AE, AB = AD and ∠BAD = ∠EAC

BC = DE

Proof,

∠BAD = ∠EAC

By Adding ∠DAC both sides

∠BAD + ∠DAC = ∠EAC + ∠DAC ∠BAC = ∠EAD

In ΔABC and ΔADE,

AC = AE (Given)

∠BAC = ∠EAD AB = AD (Given)

By SAS (Side -Angle-Side) congruence condition.

So, ΔABC ≅ ΔADE

By CPCT

BC = DE

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠BAD = ∠ABE and ∠EPA = ∠DPB (see below figure). Show that

(i) ΔDAP ≅ ΔEBP

(ii) AD = BE

Given,

P is mid-point of AB.

∠BAD = ∠ABE and ∠EPA = ∠DPB

(i) ∠EPA = ∠DPB

By Adding ∠DPE both sides

∠EPA + ∠DPE = ∠DPB + ∠DPE ∠DPA = ∠EPB

In ΔDAP ≅ ΔEBP,

∠DPA = ∠EPB

AP = BP (P is mid-point of AB)

∠BAD = ∠ABE (Given)

By ASA congruence condition. So, ΔDAP ≅ ΔEBP

(ii) By CPCT

AD = BE

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 below figure). Show that:

(i) ΔAMC ≅ ΔBMD

(ii) ∠DBC is a right angle.

(iii) ΔDBC ≅ ΔACB

(iv) CM = 1/2 AB

Given,

∠C = 90°, M is the mid-point of AB and DM = CM

(i) In ΔAMC and ΔBMD,

AM = BM (M is the mid-point)

∠CMA = ∠DMB (Vertically opposite angles)

CM = DM (Given)

By SAS(Side-Angle-Side) congruence condition

So, ΔAMC ≅ ΔBMD.

(ii)By CPCT

∠ACM = ∠BDM

Therefore, AC || BD as alternate interior angles are equal.

Now,

∠ACB + ∠DBC = 180° (co-interiors angles)

90° + ∠B = 180°

∠DBC = 90°

(iii) In ΔDBC and ΔACB,

BC = CB (Common)

∠ACB = ∠DBC (Right angles)

DB = AC (by CPCT, already proved)

By (Side-Angle-Side) congruence condition.

So, ΔDBC ≅ ΔACB

(iv) DC = AB (ΔDBC ≅ ΔACB)

DM = CM = AM = BM (M is mid-point)

DM + CM = AM + BM CM + CM = AB

CM = 1/2AB

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Class 9 Maths Class 9 Science