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# NCERT Solutions for Class 9 Maths Exercise 7.1

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NCERT solutions for Class 9 Maths Triangles

## NCERT Solutions for Class 9 Mathematics Triangles

###### 1. In quadrilateral ABCD (See figure). AC = AD and AB bisects A. Show that ABC ABD. What can you say about BC and BD?

Ans. Given: In quadrilateral ABCD, AC = AD and AB bisects A.

To prove: ABC ABD

Proof: In ABC and ABD,

BAC = BAD [ AB bisects A]

AB = AB [Common]

ABC ABD [By SAS congruency]

Thus BC = BD [By C.P.C.T.]

###### 2. ABCD is a quadrilateral in which AD = BC and DAB = CBA. (See figure). Prove that:

(i) ABD BAC

(ii) BD = AC

(iii) ABD = BAC

Ans. (i) In ABC and ABD,

DAB = CBA [Given]

AB = AB [Common]

ABC ABD [By SAS congruency]

Thus AC = BD [By C.P.C.T.]

(ii) Since ABC ABD

AC = BD [By C.P.C.T.]

(iii) Since ABC ABD

ABD = BAC [By C.P.C.T.]

###### 3. AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB (See figure)

Ans. In BOC and AOD,

BOC = AOD [Vertically Opposite angles]

BOC AOD [By ASA congruency]

OB = OA and OC = OD [By C.P.C.T.]

NCERT Solutions for Class 9 Maths Exercise 7.1

###### 4. and are two parallel lines intersected by another pair of parallel lines and (See figure). Show that ABC CDA.

Ans. AC being a transversal. [Given]

Therefore DAC = ACB [Alternate angles]

Now [Given]

And AC being a transversal. [Given]

Therefore BAC = ACD [Alternate angles]

ACB = DAC [Proved above]

BAC = ACD [Proved above]

AC = AC [Common]

ABC CDA [By ASA congruency]

NCERT Solutions for Class 9 Maths Exercise 7.1

###### 5. Line is the bisector of the angle A and B is any point on BP and BQ are perpendiculars from B to the arms of A. Show that:

(i) APB AQB

(ii) BP = BQ or P is equidistant from the arms of A (See figure).

Ans. Given: Line bisects A.

BAP = BAQ

(i) In ABP and ABQ,

BAP = BAQ [Given]

BPA = BQA = [Given]

AB = AB [Common]

APB AQB [By ASA congruency]

(ii) Since APB AQB

BP = BQ [By C.P.C.T.]

B is equidistant from the arms of A.

NCERT Solutions for Class 9 Maths Exercise 7.1

###### 6. In figure, AC = AB, AB = AD and BAD = EAC. Show that BC = DE.^

Ans. Given that BAD = EAC

Adding DAC on both sides, we get

BAD + DAC = EAC + DAC

Now in ABC and AED,

AC = AE [Given]

BAC = DAE [From eq. (i)]

BC = DE [By C.P.C.T.]

NCERT Solutions for Class 9 Maths Exercise 7.1

###### 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) DAF FBP

(ii) AD = BE (See figure)

Ans. Given that EPA = DPB

Adding EPD on both sides, we get

EPA + EPD = DPB + EPD

APD = BPE ……….(i)

Now in APD and BPE,

AP = PB [P is the mid-point of AB]

APD = BPE [From eq. (i)]

DPA EBP [By ASA congruency]

AD = BE [ By C.P.C.T.]

NCERT Solutions for Class 9 Maths Exercise 7.1

###### Show that:

(i) AMC BMD

(ii) DBC is a right angle.

(iii) DBC ACB

(iv) CM = AB

Ans. (i) In AMC and BMD,

AM = BM [AB is the mid-point of AB]

AMC = BMD [Vertically opposite angles]

CM = DM [Given]

AMC BMD [By SAS congruency]

ACM = BDM ……….(i)

CAM = DBM and AC = BD [By C.P.C.T.]

###### (ii) For two lines AC and DB and transversal DC, we have,

ACD = BDC [Alternate angles]

AC DB

Now for parallel lines AC and DB and for transversal BC.

DBC = ACB [Alternate angles] ……….(ii)

But ABC is a right angled triangle, right angled at C.

ACB = ……….(iii)

Therefore DBC = [Using eq. (ii) and (iii)]

DBC is a right angle.

(iii) Now in DBC and ABC,

DB = AC [Proved in part (i)]

DBC = ACB = [Proved in part (ii)]

BC = BC [Common]

DBC ACB [By SAS congruency]

(iv) Since DBC ACB [Proved above]

DC = AB

AM + CM = AB

CM + CM = AB [ DM = CM]

2CM = AB

CM = AB

## NCERT Solutions for Class 9 Maths Exercise 7.1

NCERT Solutions for Class 9 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 9 Maths includes text book solutions from Mathematics Book. NCERT Solutions for CBSE Class 9 Maths have total 15 chapters. 9 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 9 solutions PDF and Maths ncert class 9 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide.

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