## Basic Concepts in Geometry

### Class-9-Mathematics-2-Chapter-1-Maharashtra Board

### Notes

**Practice set 1.1 :**

**Question 1. Find the distances with the help of the number line given below :**

**(i) d(B, E) **

The distance between the two points is obtained by subtracting the smaller coordinate from the larger coordinate.

**To find d(B, E) :**

2 is a coordinate of point B

5 is a coordinate of point E.

5 > 2

###### ∴ *d*(B, E) = 5 – 2 = 3

**Answer is : d(B, E) = 3.**

** **

**(ii) d(J, A) **

The distance between the two points is obtained by subtracting the smaller coordinate from the larger coordinate.

**To find d(J, A) :**

—2 is a coordinate of point J

1 is a coordinate of point A.

1 > —2

###### ∴ *d(*J, A) = 1 – (—2) = 1 + 2 = 3

**Answer is : ***d(***J, A) ****= 3.**

** **

**(iii) d(P, C) **

The distance between the two points is obtained by subtracting the smaller coordinate from the larger coordinate.

**To find d(P, C) :**

—4 is a coordinate of point P

3 is a coordinate of point C.

3 > —4

###### ∴ *d(*P, C) = 3 – (—4) = 3 + 4 = 7

**Answer is : ***d(***P, C) ****= 7.**

** **

**(iv) d(J, H)**

**To find d(J, H) :**

—2 is a coordinate of point J

—1 is a coordinate of point H.

—1 > —2

###### ∴ *d(*J, H) = —1 – (—2) = —3 + 2 = 1

**Answer is : ***d(***J, H) ****= 1.**

** **

**(v) d(K, O) **

**To find d(K, O) :**

—3 is a coordinate of point K

0 is a coordinate of point O.

0 > —3

###### ∴ *d(*K, O) = 0 – (—3) = 0 + 3 = 3

**Answer is : ***d(***K, O) ****= 3.**

** **

**(vi) d(O, E) **

**To find d(O, E) :**

0 is a coordinate of point O

5 is a coordinate of point E.

5 > 0

###### ∴ *d(*O, E) = 5 – 0 = 5

**Answer is : ***d(***O, E) ****= 5.**

**(vii) d(P, J) **

** To find d(P, J) **

—4 is a coordinate of point P

—2 is a coordinate of point J.

—2 > —4

###### ∴ *d(*P, J) = —2 – (—4) = —2 + 4 = 2

**Answer is : ***d(***P, J) ****= 2.**

**(viii) d(Q, B)**

**To find d(Q, B)**

—5 is a coordinate of point Q

2 is a coordinate of point B.

2 > —5

∴ *d(*Q, B) = 2 – (—5) = 2 + 5 = 7

**Answer is : ***d(***Q, B) ****= 7.**

**Question ****2. If the co—ordinate of A is x and that of B is y**

**, find**

*d(***A, B) .**

**(i) x = 1, y = 7 **

Coordinate of point A is x = 1.

###### Coordinate of point Bis y = 7

###### 7 > 1

∴ d(AB) = 7 – 1 = 6**Answer is : ****d(AB) = 6**

**(ii) x = 6, y = **

**—**

**2**

Coordinate of point A is x = 6.

###### Coordinate of point B is y = —2

###### 6 > —2

∴ d(AB) = 6 – (—2) = 6 + 2 = 8**Answer is : ****d(AB) = 8**

**(iii) x = **

**—**

**3,**

*y*= 7

Coordinate of point A is x = —3

###### Coordinate of point B is y = 7

###### 7 > —3

∴ d(AB) = 7 – (—3) = 7 + 3 = 10**Answer is : ****d(AB) = 10**

**(iv) x = **

**—**

**4,**

*y*=**—**

**5**

Coordinate of point A is x = —4

###### Coordinate of point B is y = —5

###### —4 > —5

∴ d(AB) = —4 – (—5) = —4 + 5 = 1**Answer is : ****d(AB) = 1**

**(v) x = **

**—**

**3,**

*y*=**—**

**6**

Coordinate of point A is x = —3

###### Coordinate of point B is y = —6

###### —3 > —6

∴ d(AB) = —3 – (—6) = —3 + 6 = 3**Answer is : ****d(AB) = 3**

**(vi) x = 4, y = **

**—**

**8**

Coordinate of point A is x = 4

###### Coordinate of point B is y = —8

###### 4 > —8

∴ d(AB) = 4 – (—8) = 4 + 8 = 12**Answer is : ****d(AB) = 12**

**Question ****3. ****From the information given below, find which of the point is between the other two. ****If the points are not collinear, state so.**

**(i) d(P, R) = 7, d(P, Q) = 10, d(Q, R) = 3**

*d(*P, R) = 7, *d(*P, Q) = 10, *d(*Q, R) = 3 ….(given)

d(P, R) + d(Q, R) = 7 + 3 = 10

d(P, Q) = 10

∴ d(P, R) + d(Q, R) = d(P, Q)

∴ points P, Q and R are collinear.

Point R lies between points P and Q.

**Answer is : P****—****R****—****Q or Q****—****R****—****P.**

**(ii) d(R, S) = 8, d(S, T) = 6, d(R, T) = 4**

*d(*R, S) = 8, *d(*S, T) = 6, *d(*R, T) = 4 ….(given)

d(S, T) + d(R, T) = 6 + 4 = 8

d(R, S) = 8

∴ d(S, T) + d(R, T) ≠ d(R, S)

∴ points R, S and T are not collinear.

**Answer is : Points R, S and T are not collinear.**

**(iii) d(A, B) = 16, d(C, A) = 9, d(B, C) = 7**

*d(*A, B) = 16, *d(*C, A) = 9, *d(*B, C) = 7 ….(given)

d(C, A) + d(B, C) = 9 + 7 = 16

d(A, B) = 16

∴ d(C, A) + d(B, C) = d(A, B)

∴ points A, B and C are collinear.

Point C lies between points A and B.

**Answer is : A****—****C****—B**** or B****—****C****—A****.**

**(iv) d(L, M) = 11, d(M, N) = 12, d(N, L) = 8**

*d(*L, M) = 11, *d(*M, N) = 12, *d(*N, L) = 8** **….(given)

d(L, M) + d(N, L) = 11 + 8 = 19

d(M, N) = 12

∴ d(L, M) + d(N, L) ≠ d(M, N)

∴ points L, M and N are not collinear.

**Answer is : Points L, M and N are not collinear.**

**(v) d(X, Y) = 15, d(Y, Z) = 7, d(X, Z) = 8**

*d(*A, B) = 16, *d(*C, A) = 9, *d(*B, C) = 7 ….(given)

d(C, A) + d(B, C) = 9 + 7 = 16

d(A, B) = 16

∴ d(C, A) + d(B, C) = d(A, B)

∴ points A, B and C are collinear.

Point C lies between points A and B.

**Answer is : A****—****C****—B**** or B****—****C****—A****.**

**(vi) d(D, E) = 5, d(E, F) = 8, d(D, F) = 6**

*d*(D, E) = 5, *d*(E, F) = 8, *d*(D, F) = 6** **….(given)

d(D, E) + d(D, F) = 5 + 6 = 11

d(E, F) = 8

∴ d(D, E) + d(D, F) ≠ d(E, F)

∴ points D, E and F are not collinear.

**Answer is : Points D, E and F are not collinear.**

**Question ****4. On a number line, points A, B and C are such that d(A, C) = 10, d(C, B) = 8. Find d(A, B) considering all possibilities.**

*d(*A, C) = 10, *d(*C, B) = 8 ** **….(given)

**Possibility 1 :**

C lies between A and B, i.e. A—C —B.

d(A, B) = d(A, C) + d(C, B) ..(A—C—B)

∴ d(A, B) = 10 + 8

∴ d(A, B) = 18

When C lies between A and B, d(A, B) = 18

**Possibility 2 :**

B lies between A and C, i.e. A—B —C.

d(A, B) + d(B, C) = d(A, C)

∴ d(A, B) + 8 = 10

###### ∴ d(A, B) = 10 — 8

∴ d(A, B) = 2

When B lies between A and C, d(A, B) =2

**Answer is : d(A, B) =18 or 2.**

**Question ****5. Points X, Y, Z are collinear such that d(X, Y) = 17, d(**

**Y, Z) = 8, find**

*d(***X, Z) .**

d(X, Y) = 17; d(Y, Z) = 8 …..(Given)

Points X, Y and Z are collinear.

Possibility 1 :

Point Z lies between points X and Y i.e. X—Z—Y.

d(X, Z) + d(Z, Y) = d(X, Y) ….(X—Z—Y)

∴ d(X, Z) + 8 = 17

∴ d(X, Z) = 17 — 8

∴ d(X, Z) = 9

**Possibility 2 :**

Point Y lies between points X and Z i.e. X—Y—Z.

d(X, Z) = d(X, Y) + d(Y, Z) ….(X—Y—Z)

∴ d(X, Z) = 17 + 8

∴ d(X, Z) = 25

**Answer is : d(X, Z) = 25 or 9.**

**Question ****6. ****Sketch proper figure and write the answers of the following questions.**

**(i) If A—B—C and l(AC) = 11, l(BC) = 6.5, then l(AB) =?**

*l*(AC) = 11, *l*(BC) = 6.5

*l*(AB) + *l*(BC) = *l*(AC) …..(A—B—C)

∴ *l*(AB) + 6.5 = 11

∴ *l*(AB) = 11 — 6.5

∴ *l*(AB) = 4.5

**Answer is : l(AB) = 4.5.**

**(ii) If R—S—T and l(ST) = 3.7, l(RS) = 2.5, then l(RT) =?**

*l*(RS) = 2.5, *l*(ST) = 3.7

*l*(RS) + *l*(ST) = *l*(RT) ...(R—S—T)

∴ 2.5 + 3.7 = *l*(RT)

∴ *l*(RT) = 6.2

**Answer is : l(RT) = 6.2**

**(iii) If X—Y—Z and l(XZ) = **

**3**

**,**

*l*(XY) =

**, then**

*l*(YZ) = ?

*l*(XY) = \(\sqrt{7}\), *l*(XZ) = 3\(\sqrt{7}\)

*l*(XY) + *l*(YZ) = *l*(XZ) ……(X—Y~Z)

∴ \(\sqrt{7}\) + *l*(YZ) = 3\(\sqrt{7}\)

∴ *l*(YZ) = 3\(\sqrt{7}-\sqrt{7}\)

∴ *l*(YZ) = 2\(\sqrt{7}\)

**Answer is : l(YZ) = 2\(\sqrt{7}\)**

**Question ****7. ****Which figure is formed by three non—collinear points ****?**

A triangle is formed by three non—collinear points.

**Practice set 1.2 :**

**Question 1. The following table shows points on a number line and their co—ordinates. Decide whether the pair of segments given below the table are congruent or not.**

Point |
A |
B |
C |
D |
E |

Co—ordinate |
—3 |
5 |
2 |
—7 |
9 |

** ****(i) seg DE and seg AB **

**seg DE :**

*l*(DE) = d(D, E)

Coordinate of point D is — 7

###### Coordinate of point E is 9

9 > — 7

∴ d(D, E) = 9 — ( — 7)_

∴ d(D,E) = 9 + 7,

∴ d(D,E) = 16

∴ *l*(DE) = 16 …..(1)

**seg AB :**

*l*(AB) = d(A, B)

Coordinate of pointA is —3,

Coordinate of point B is 5

5 > —3

∴ d(A,B)=5—(—3)

∴ d(A, B) = 5 + 3

∴ d(A,B) = 8

∴ *l*(AB) = 8 ……(2)

∴ *l*(DE) ≠ *l*(AB) ……[From (1) and (2)]

**Answer is : seg DE and seg AB are not congruent.**

**(ii) seg BC and seg AD **

**seg BC :**

*l*(BC) = d(B, C)

Coordinate of point B is 5

Coordinate of point C is 2

5 > 2

∴ d(B, C) = 5 — 2

∴ d(B, C) = 3

∴ *l*(BC) = 3 …....(1)

**seg AD :**

*l*(AD) = d(A, D)

Coordinate of point A is —3

###### Coordinate of point D is —7

—3 > —7

∴ d(A,D) = —3 — (—7)

∴ d(A,D) = —3 + 7

∴ d(A, D) = 4

*l*(AD) = 4 ……..(2)

*l*(BC) ≠ *l*(AD) ……[From (1) and (2)]

**Answer is : seg BC is not congruent to seg AD.**

**(iii) seg BE and seg AD**

**seg BE :**

*l*(BE) = d(B, E)

Coordinate of point B is 5

Coordinate ofpoint E is 9

9 > 5

∴ d(B, E) = 9 — 5

∴ d(B, E) = 4

∴ *l*(BE) = 4 ...(1)

**seg AD :**

l(AD) = d(A, D)

Coordinate ofp0intA is — 3

Coordinate ofpoint D is —7

—3 > —7

d(A,D) = —3 —(—7)

∴ d(A,D) = —3 + 7

∴ d(A, D) = 4

∴ *l*(AD) = 4 …..(2)

* l*(BE) = *l*(AD) …..[From (1) and (2)]

seg BE ≅ seg AD

**Answer is : seg BE and seg AD are congruent.**

**Question 2. ****Point M is the midpoint of seg AB. If AB = 8 then find the length of AM.**

*l*(AB) = 8 ……(Given)

*l*(AM) = \(\frac{1}{2}\) *l*(AB) …..(M is the midpoint of seg AB)

∴ *l*(AM) = \(\frac{1}{2}\) × 8

∴* l*(AM) = 4

**Answer is : l(AM) = 4**

**Question 3. ****Point P is the midpoint of seg CD. If CP = 2.5, find ***l***(CD).**

*l*(CP) = 2.5 …..(Given)

*l*(CD) = 2*l*(CP) …..(P is the midpoint of seg CD)

∴ *l*(CD) = 2 × 2.5

∴ *l*(CD) = 5

**Answer is : l(CD) = 5.**

**Question 4. If AB = 5 cm, BP = 2 cm and AP = 3.4 cm, compare the segments.**

**Given : **AB = 5 cm, BP = 2 cm, AP = 3.4 cm

5 > 3.4 > 2

∴ AB > AP > BP

∴ segAB > segAP > scg BP.

**Question 5. ****Write the answers to the following questions with reference to figure**

**(i) Write the name of the opposite ray of ray RP**

Ray RS or Ray RT is the ray opposite to ray RP.

**(ii) Write the intersection set of ray PQ and ray RP.**

Ray PQ is the intersection set of ray PQ and ray RP.

**(iii) Write the union set of seg PQ and seg QR.**

Line PQ or line QR is the union set of ray PQ and ray QR.

**(iv) State the rays of which seg QR is a subset.**

Seg QR is a subset of ray RQ, ray SQ, ray TQ, ray QR, ray QS, ray QT.

**(v) Write the pair of opposite rays with common end point R.**

Ray RP and ray RS is the pair of opposite rays with common endpoint R.

**(vi) Write any two rays with common end point S.**

Ray SR and ray ST are two rays with common endpoint S.

**(vii) Write the intersection set of ray SP and ray ST****.**

The intersection of ray SP and ray ST is point S.

**Question 6. ****Answer the questions with the help of figure.**

**(i) State the ****points which are equidistant from point B.**

- Point C and point A are equidistant from point B as both of the points are at a distance of 2 units from B.

###### · Also, Point D and point P are equidistant from point B as both of the points are at a distance of 4 units from B.

**(ii) Write a pair of points equidistant from point Q.**

- Point L and point U is the pair of points equidistant from point Q as both of the points are at a distance of 1 unit from point Q.
- Also, Point P and point R is the pair of points equidistant from point Q as both of the points are at a distance of 2 units from point Q.

**(iii) Find d(U,V), d(P,C), d(V,B), d**

**(U, L).**

**For d(U, V):**

Coordinate of point U is — 5

Coordinate of point V is 5

5 > —5

d(U, V) = 5 - (—5)

d(U, V) = 5 + 5

d(U, V) = 10

**Answer is : d(U, V) = 10.**

**For d(P, C) :**

Coordinate of point P is —2

Coordinate of point C is 4

4 > — 2

d(P, C) = 4—(—2)

d(P, C) = 4 + 2

d(P, C) = 6

**Answer is : d(P, C) = 6.**

**For d(V, B) :**

Coordinate of point V is 5

Coordinate of point B is 2

5 > 2

a’(V, B)=5 —2

d(V, B) = 3

**Answer is : d(V, B) = 3**

**For d(U, L) :**

Coordinate of point U is —5

Coordinate of point L is —3

—3 > —5

d(U, L) = -3 -(-5)

ﬂ'(U,L) = -3 + 5

d(U, L) = 2

**Answer is : d(U, L) = 2.**

**Practice set 1.3 :**

**Question 1. Write the following statements in ‘if -then’ form :**

**(i) The opposite angles of a parallelogram are congruent.**

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

**(ii) The diagonals of a rectangle are congruent.**

If a quadrilateral is a rectangle, then its diagonals are congruent.

** **

**(iii) In an isosceles triangle, the segment joining the vertex and the midpoint of the haw is perpendicular to the base.**

If a triangle is an isosceles triangle, then the segment joining the vertex and the midpoint of the base is perpendicular to the base.

**Question 2. Write converses of the following statements :**

**(i) The alternate angles formed by two parallel lines and their transversal are congruent.**

If alternate angles made by two lines and its transversal are congruent, then the lines are parallel.

**(ii) lf a pair of the interior angles made by a transversal of two lines are supplementary then the lines are parallel.**

If two parallel lines are intersected by a transversal, then the interior angles so formed are supplementary.

**(iii) The diagonals of a rectangle are congruent.**

If the diagonals of a quadrilateral are congruent, then that quadrilateral is a rectangle.

**Problem set 1 :**

**Question ****1. Select the correct alternative from the answers of the questions given below.**

**(i) How many mid points does a segment have ?**

**(A) only one (B) two (C) three (D) many**

(A) only one

**(ii) How many points are there in the intersection of two distinct lines ?**

**(A) infinite (B) two (C) one (D) not a single**

(C) one

**(iii) How many lines are determined by three distinct points****?**

**(A) two (B) three (C) one or three (D) six**

(C) one or three

**(iv) Find d(A, B), if co-ordinates of A and B are ****- ****2 and 5 respectively.**

**(A) ****-****2 (B) 5 (C) 7 (D) 3**

(C) 7

**(v) If P-Q-R and d(P, Q) = 2, d**

**(P, R) = 10, then find**

*d***(Q, R).**

**(A) 12 (B) 8 (C) ****96 ****(D) 20**

(B) 8

**Question ****2. On a number line, co-ordinates of P, Q, R are 3, ****- ****5 and 6 respectively. State with reason whether the following statements are true or false.**

**(i) d(P, Q) + d(Q, R) = d(P, R) **

**(ii) d(P, R) + d(R, Q) = d(P, Q)**

**(iii) d(R, P) + d(P, Q) = d(R, Q) **

** (iv) d(P, Q) **

**-**

*d***(P, R) =**

*d*(Q, R)

**Given : **Coordinate of point Q is -5, Coordinate of point P is 3, Coordinate of point R is 6

**Find d(P, Q) :**

Coordinate of point P is 3

Coordinate of point Q is —5

3 > —5

∴ d(P, Q) = 3—(—5)

∴ d(P, Q) = 3 + 5

∴ d(P, Q) = 8

**Find d(Q, R) :**

Coordinate of point Q is —5

Coordinate of point R is 6

6 > — 5

∴ d(Q, R) = 6 — (—5)

∴ d(Q, R) = 6 + 5

∴ d(Q, R) = 11

**Find d(P, R) :**

Coordinate of point P is 3

Coordinate of point R is 6

6 > 3

∴ d(P, R) = 6—3

∴ d(P, R) = 3

**(i) **d(P, Q) + d(Q, R) = 8 + 11

∴ d(P, Q) + d(Q, R) = 19

d(P, R) = 3

∴ d(P, Q) + d(Q, R) ≠ d(P, R)

Answer is : d(P, Q) + d(Q, R) =d(P, R) is a false statement.

**(ii) **d(P, R) + d(R, Q) = 3 + 11

∴ d(P, R) + d(R, Q) = 14

d(P, Q) = 8

∴ d(P, R) + d(R, Q) ≠ d(P, Q)

**Answer is : d(P, R) + d(R, Q) = d(P, Q) is a false statement.**

**(iii) **d(R, P) + d(P, Q) = 3 + 8

∴ d(R, P) + d(P, Q) = 11

d(R, Q) = 11

∴ d(R, P) + d(P, Q) = d(R, Q)

**Answer is : d(R, P) + d(P, Q) = d(R, Q) is a true statement. **

**(iv) **d(P, Q) — d(P, R) = 8 — 3

∴ d(P, Q) — d(P, R) = 5

d(Q, R) = 11

∴ d(P, Q) — d(P, R) ≠ d(Q, R)

**Answer is : d(P, Q) — d(P, R) =d(Q, R) is a false statement.**

**Question ****3. Co—ordinates of some pairs o****f points are given below. Hence find the distance between ****each pair.**

**(i) 3, 6 **

###### Coordinate of the ﬁrst point = 3

Coordinate of the second point = 6

6 > 3

The distance between the two points

= 6 —3 = 3

**Answer is : Distance between the two points is 3.**

**(ii) ****— ****9, ****— ****1 **

###### Coordinate of the ﬁrst point = —9

Coordinate of the second point = —1

—1 > —9

The distance between the two points

= —1 — (—9)

= — 1 + 9 = 8

**Answer is : Distance between the two points is 8**

**(iii) ****— ****4, 5 **

Coordinate of the ﬁrst point = —4

Coordinate of the second point = 5

5 >—4

The distance between the two points

= 5 — (—4) = 5 + 4 = 9

Answer is: Distance between the two points is 9

**(iv) 0 , **

**—**

**2**

Coordinate of the ﬁrst point = 0

Coordinate of the second point = —2

0 > —2

The distance between the two points

= 0 — (—2) = 0 + 2 = 2

**Answer is : Distance between the two points is 2**

**(v) x + 3, x **

**—**

**3**

Coordinate of the ﬁrst point is x + 3

Coordinate of the second point is x — 3

For any real value of x, x + 3 > x — 3

x + 3 > x — 3

The distance between the two points

= (x + 3) — (x — 3)

= x + 3 — x + 3 = 6

**Answer is : Distance between the two points is 6.**

**(vi) ****— ****25, ****— ****47 **

Coordinate of the ﬁrst point = —25

Coordinate of the second point = —47

— 25 > — 47

The distance between the two points

= —25 — (—47) = —25 + 47 = 22

**Answer is : Distance between the two points is 22**

**(vii) 80, ****— ****85**

Coordinate of the ﬁrst point = 80

Coordinate of the second point = —85

80 > —85

The distance between the two points

= 80 — (—85) = 80 + 85 = l65

**Answers is: Distance between the two points is 165**

**Question ****4. Co—ordinate of point P on a number line is ****— ****7. Find the co—ordinates of points on the number line which are at a distance of 8 units from point P.**

Let A, having coordinate x, be the point on the left side of P at a distance of 8 from P.

Coordinate of point P is — 7.

— 7 > x

d(A, P) = 8

∴ 8 = —7 — x

∴ x = —7 — 8

x = — 15

Coordinate of point A is — 15.

Let B, having coordinate y, be the point on the right side of P at a distance of 8 from P.

Y > —7,

d(P, B) = 8

∴ 8 = y — (—7)

∴ 8 = y + 7

∴ y = 8 — 7

y = 1

Coordinate of point B is 1.

**Ans. Coordinates of points on the number line which are at a distance of 8 units from P are —15 and 1.**

**Question ****5. Answer the following questions.**

**(i) If A — B — C and d**

**(A,C) = 17,**

*d***(B,C) = 6.5 then**

*d*(A,B) = ?

d(A, B) + d(B, C) = d(A, C) …..(A — B — C)

d(A, B) + 6.5 = 17

d(A, B) = 17 — 6.5

∴ d(A, B) = 10.5

**Answer is: d(A, B) = 10.5.**

**(ii) If P — Q — R and d(P,Q) = 3.4, d(Q,R)= 5.7 then d(P,R) = ?**

d(P, R) = d(P, Q) + d(Q, R) ….( P — Q – R)

d(P, R) = 3.4 + 5.7

d(P, R) = 9.1

**Answer is: d(P, R) = 9.1**

**Question ****6. ****Co—ordinate of point A on a number line is 1. What are the co—ordinates of points on the ****number line which are at a distance of 7 units from A ?**

Coordinate of point A is 1.

Let B, having coordinate x, be the point on the left side ofA at a distance of 7 from A.

1 > x, d(A, B) = 7

∴ 7 = 1—x

∴ x = 1 —7 = —6

Coordinate of point B is — 6.

Let C having coordinate y, be the point on the right side of A at a distance of 7 from A.

y > 1, d(A, C) = 7

∴ 7 = y — 1.

∴ y = 7 + l = 8

Coordinate of point C is 8.

**Ans. Coordinates of points on number line which are at a distance of 7 units from A are — 6 and 8.**

**Question ****7. ****Write the following statements in conditional form.**

**(i) Every rhombus is a square.**

If the quadrilateral is a rhombus, then it is a square.

**(ii) Angles in a linear pair are supplementary.**

If the angles are in a linear pair, then they are supplementary.

**(iii) A triangle is a figure formed by three segments.**

If the ﬁgure is a triangle, then it is formed by three segments.

**(iv) A number having only two divisors is called a prime number.**

If a number is having only two divisors, then it is called a prime number.

**Question ****8. ****Write the converse of each of the following statements.**

**(i) If the sum of measures of angles in a figure is 1800, then the figure is a triangle.**

If the ﬁgure is a triangle, then the sum of the measures of the angles is 180°.

**(ii) If the sum of measures of two angles is 900 ****then they are complement of each other.**

If two angles are complement of each other, then the sum of the measures of two angles is 90°.

**(iii) If the corresponding angles formed by a transversal of two lines are congruent then the two lines are parallel.**

If the two lines are parallel, then the corresponding angles formed by a transversal of two lines are congruent.

**(iv) If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.**

If the number is divisible by 3, then the sum of the digits of the number is also divisible by 3.

**Question ****9. ****Write the antecedent (given part) and the consequent (part to be proved) in the following ****statements.**

**(i) If all sides of a triangle are congruent then its all angles are congruent.**

**Antecedent : **All sides of a triangle are congruent.

**Consequent : **Its all angles are congruent.

**(ii) The diagonals of a parallelogram bisect each other.**

Consider statement in conditional form. ‘If a quadrilateral is a parallelogram, then its diagonals bisect each other.

**Antecedent :** Quadrilateral is a parallelogram.

**Consequent** : Diagonals bisect each other.

**Question ****10*********. Draw a labelled figure showing information in each of the following statements and ****write the antecedent and the consequent.**

**(i) Two equilateral triangles are similar.**

Consider statement in conditional form ‘lf two triangles are equilateral, then they are similar.’

**Antecedent** : Two triangles are equilateral. i.e. Δ ABC and Δ PQR are equilateral.

**Consequent :** Triangles are similar.

i.e. Δ ABC ~ Δ PQR.

**(ii) If angles in a linear pair are congruent then each of them is a right angle.**

**Antecedent **: Angles in a linear pair arc congruent. i.e. ∠ ABC ≅ ∠CBD

**Consequent** : Each of the angles is a right angle.

i.e. ∠ ABC = ∠ CBD = 90°.

**(iii) If the altitudes drawn on two sides of a triangle are congruent then those two sides are congruent.**

**Antecedent **: Altitudes drawn on two sides of a triangle are congruent.

i.e. in Δ ABC,

seg BM ⊥ side AC, A—M—C

seg CN ⊥ side AB, A—N—B

seg BM ≅ seg CN.

**Consequent** : Those two sides are congruent.

i.e. side AB ≅ side AC.

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