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#### Question 1:

Find the distances with the help of the number line given below.

(i) d(B,E) (ii) d(J, A) (iii) d(P, C)   (iv) d(J, H)  (v) d(K, O)

(vi) d(O, E) (vii) d(P, J) (viii) d(Q, B)

It is known that, distance between the two points is obtained by subtracting the smaller co-ordinate from larger co-ordinate.

(i) The co-ordinates of points B and E are 2 and 5 respectively. We know that 5 > 2.
d(B, E) = 5 − 2 = 3

(ii) The co-ordinates of points J and A are −2 and 1 respectively. We know that 1 > −2.
∴ d(J, A) = 1 − (−2) = 1 + 2 = 3

(iii) The co-ordinates of points P and C are −4 and 3 respectively. We know that 3 > −4.
∴ d(P, C) = 3 − (−4) = 3 + 4 = 7

(iv) The co-ordinates of points J and H are −2 and −1 respectively. We know that −1 > −2.
∴ d(J, H) = −1 − (−2) = −1 + 2 = 1

(v) The co-ordinates of points K and O are −3 and 0 respectively. We know that 0 > −3.
∴ d(K, O) = 0 − (−3) = 0 + 3 = 3

(vi) The co-ordinates of points O and E are 0 and 5 respectively. We know that 5 > 0.
∴ d(O, E) = 5 − 0 = 5

(vii) The co-ordinates of points P and J are −4 and −2 respectively. We know that −2 > −4.
∴ d(P, J) = −2 − (−4) = −2 + 4 = 2

(viii) The co-ordinates of points Q and B are −5 and 2 respectively. We know that 2 > −5.
∴ d(Q, B) = 2 − (−5) = 2 + 5 = 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               (ii) x = 6, y = $-$2           (iii) x $-$3, y = 7

(iv) x$-$4, y $-$5      (v) x = $-$3, y $-$6 (vi) x = 4, y = $-$8

It is known that, distance between the two points is obtained by subtracting the smaller co-ordinate from larger co-ordinate.

(i) The coordinates of A and B are x and y respectively. We have, x = 1 and y = 7. We know that 7 > 1.
d(A, B) = y − x = 7 − 1 = 6

(ii) The coordinates of A and B are x and y respectively. We have, x = 6 and y = −2. We know that 6 > −2.
∴ d(A, B) = x − y = 6 − (−2) = 6 + 2 = 8

(iii) The coordinates of A and B are x and y respectively. We have, x = −3 and y = 7. We know that 7 > −3.
∴ d(A, B) = y − x = 7 − (−3) = 7 + 3 = 10

(iv) The coordinates of A and B are x and y respectively. We have, x = −4 and y = −5. We know that −4 > −5.
∴ d(A, B) = x − y = −4 − (−5) = −4 + 5 = 1

(v) The coordinates of A and B are x and y respectively. We have, x = −3 and y = −6. We know that −3 > −6.
∴ d(A, B) = x − y = −3 − (−6) = −3 + 6 = 3

(vi) The coordinates of A and B are x and y respectively. We have, x = 4 and y = −8. We know that 4 > −8.
∴ d(A, B) = x − y = 4 − (−8) = 4 + 8 = 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

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

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

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

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

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

(i) We have, d(P, R) = 7; d(P, Q) = 10; d(Q, R) = 3
Now, d(P, R) + d(Q, R) = 7 + 3
Or, d(P, R) + d(R, Q) = 10
∴ d(P, Q) = d(P, R) + d(Q, R)
Hence, the points P, R and Q are collinear.
The point R is between P and Q i.e., P-R-Q.

(ii) We have, d(R, S) = 8; d(S, T) = 6; d(R, T) = 4
Now, 8 + 6 = 14, so 8 + 6 ≠ 4; 6 + 4 = 10, so 6 + 4 ≠ 8 and 8 + 4 = 12, so 8 + 4 ​≠ 6
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points R, S and T are non-collinear.

(iii) We have, d(A, B) = 16; d(C, A) = 9; d(B, C) = 7
Now, d(C, A) + d(B, C) = 9 + 7
Or, d(A, C) + d(C, B) = 16
∴ d(A, B) = d(A, C) + d(C, B)
Hence, the points A, C and B are collinear.
The point C is between A and B i.e., A-C-B.

(iv) We have, d(L, M) = 11; d(M, N) = 12; d(N, L) = 8
Now, 11 + 12 = 23, so 11 + 12 ≠ 8; 12 + 8 = 20, so 12 + 8 ≠ 11 and 11 + 8 = 19, so 11 + 8 ​≠ 12
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points L, M and N are non-collinear.

(v) We have, d(X, Y) = 15; d(Y, Z) = 7; d(X, Z) = 8
Now, d(X, Z) + d(Y, Z) = 7 + 8
Or, d(X, Z) + d(Z, Y) = 15
∴ d(X, Y) = d(X, Z) + d(Z, Y)
Hence, the points X, Z and Y are collinear.
The point Z is between X and Y i.e., X-Z-Y.

(vi) We have, d(D, E) = 5,   d(E, F) = 8,   d(D, F) = 6
Now, 5 + 8 = 13, so 5 + 8 ≠ 6; 8 + 6 = 14, so 8 + 6 ≠ 5 and 5 + 6 = 11, so 5 + 6 ​≠ 8
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points D, E and F are non-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.

There are only two possibilities.
Case 1 : When point C is between the points A and B.

We have, d(A, C) = 10; d(C, B) = 8
Now, d(A, B) = d(A, C) + d(C, B) = 10 + 8
∴ d(A, B) = 18
Case 2 : When point B is between the points A and C.

We have, d(A, C) = 10; d(C, B) = 8
Now, d(A, C) = d(A, B) + d(B, C)
So, d(A, B) = d(A, C) − d(B, C) = 10 − 8
∴ d(A, B) = 2

#### Question 5:

Points X, Y, Z are collinear such that d(X,Y) = 17, d(Y,Z) = 8, find d(X,Z) .

It is given that the points X, Y and Z are collinear.
We have d(X,Y) = 17; d(Y,Z) = 8.
Now, d(X,Z) = d(X,Y) + d(Y,Z) = 17 + 8
∴ d(X,Z) = 25

#### 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) =?

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

(iii) If X - Y - Z and l(XZ) = 3 $\sqrt{7}$, l(XY) =$\sqrt{7}$ , then l(YZ) =?

(i)

We have, l(AC) = 11; l(BC) = 6.5.
Now, l(AC) = l(AB) + l(BC)
So, l(AB) = l(AC) − l(BC) = 11 − 6.5
∴ l(AB) = 4.5

(ii)

We have, l(ST) = 3.7; l(RS) = 2.5.
Now, l(RT) = l(RS) + l(ST) = 3.7 + 2.5
∴ l(RT) = 5.6

(iii)

We have, l(XZ) = $3\sqrt{7}$ l(XY) = $\sqrt{7}$.
Now, l(XZ) = l(XY) + l(YZ)
So, l(YZ) = l(XZ) −  l(XY) = $3\sqrt{7}$ − $\sqrt{7}$
∴ l(YZ) = $2\sqrt{7}$

#### Question 7:

Which figure is formed by three non-collinear points ?

A triangle is formed by three segments joining three non-collinear points.

A, B and C are three non-collinear points. When A, B and C are joined, we get a ∆ABC.

#### 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 (ii) seg BC and seg AD (iii) seg BE and seg AD

The given table is,

 Point A B C D E Co-ordinate $-$3 5 2 $-$7 9

(i) The co-ordinates of points D and E are −7 and 9 respectively. We know that 9 > −7.
∴ l(DE) = 9 − (−7) = 9 + 7 = 16
The co-ordinates of points A and B are −3 and 5 respectively. We know that 5 > −3.
∴ l(AB) = 5 − (−3) = 5 + 3 = 8
Since, l(DE) ≠ l(AB), so seg DE ≇ seg AB.

(ii) The co-ordinates of points B and C are 5 and 2 respectively. We know that 5 > 2.
∴ l(BC) = 5 − 2 = 3
The co-ordinates of points A and D are −3 and −7 respectively. We know that −3 > −7.
∴ l(AD) = −3 − (−7) = −3 + 7 = 4

(iii) The co-ordinates of points B and E are 5 and 9 respectively. We know that 9 > 5.
∴ l(BE) = 9 − 5 = 4
The co-ordinates of points A and D are −3 and −7 respectively. We know that −3 > −7.
∴ l(AD) = −3 − (−7) = −3 + 7 = 4

#### Question 2:

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

We have l(AB) = 8.
Since, M is the midpoint of seg AB, then
l(AM) = $\frac{1}{2}$ of l(AB)
∴ l(AM) = $\frac{1}{2}$ × 8 = 4
So, length of AM is 4.

#### Question 3:

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

We have l(CP) = 2.5.
Since, P is the midpoint of seg CD, then
l(CP) = $\frac{1}{2}$ of l(CD)
∴ l(CD) = 2 × l(CP) = 2 × 2.5 = 5
So, length of CD is 5.

#### Question 4:

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

We have l(AB) = 5 cm; l(BP) = 2 cm; l(AP) = 3.4 cm
We know that 5 > 3.4 > 2.
So, l(AB) > l(AP) > l(BP).
∴ seg AB > seg AP > seg BP.

#### Question 5:

Write the answers to the following questions with reference in the given figure .

(i) Write the name of the opposite ray of ray RP.
(ii) Write the intersection set of ray PQ and ray RP.
(iii) Write the union set of ray PQ and ray QR.
(iv) State the rays of which seg QR is a subset.
(v) Write the pair of opposite rays with common end point R.
(vi) Write any two rays with common end point S.
(vii) Write the intersection set of ray SP and ray ST.

(i) Ray RS or Ray RT

(ii) Ray PQ

(iii) Ray QR

(iv) Ray QR, Ray RQ etc.

(v) Ray RQ and Ray RT etc.

(vi) Ray ST and Ray SR etc.

(vii) Point S

#### Question 6:

Answer the questions with the help of  a given figure.

(i) State the points which are equidistant from point B.
(ii) Write a pair of points equidistant from point Q.
(iii) Find d(U,V), d(P,C), d(V,B),d(U, L).

(i) The co-ordinates of points B and C are 2 and 4 respectively. We know that 4 > 2.
d(B, C) = 4 − 2 = 2
The co-ordinates of points B and A are 2 and 0 respectively. We know that 2 > 0.
d(B, A) = 2 − 0 = 2
Since d(B, A) =  d(B, C), then points A and C are equidistant from point B.
The co-ordinates of points B and D are 2 and 6 respectively. We know that 6 > 2.
∴ d(B, D) = 6 − 2 = 4
The co-ordinates of points B and P are 2 and −2 respectively. We know that 2 > −2.
∴ d(B, P) = 2 − (−2) = 2 + 2 = 4
Since d(B, D) =  d(B, P), then points D and P are equidistant from point B.

(ii) The co-ordinates of points Q and U are −4 and −5 respectively. We know that −4 > −5.
∴ d(Q, U) = −4 − (−5) = −4 + 5 = 1
The co-ordinates of points Q and L are −4 and −3 respectively. We know that −3 > −4.
∴ d(Q, L) = −3 − (−4) = −3 + 4 = 1
Since d(Q, U) =  d(Q, L), then points U and L are equidistant from point Q.
The co-ordinates of points Q and R are −4 and −6 respectively. We know that −4 > −6.
∴ d(Q, R) = −4 − (−6) = −4 + 6 = 2
The co-ordinates of points Q and P are −4 and −2 respectively. We know that −2 > −4.
∴ d(Q, P) = −2 − (−4) = −2 + 4 = 2
Since d(Q, R) =  d(Q, P), then points R and P are equidistant from point Q.

(iii) The co-ordinates of points U and V are −5 and 5 respectively. We know that 5 > −5.
∴ d(U, V) = 5 − (−5) = 5 + 5 = 10
The co-ordinates of points P and C are −2 and 4 respectively. We know that 4 > −2.
∴ d(P, C) = 4 − (−2) = 4 + 2 = 6
The co-ordinates of points V and B are 5 and 2 respectively. We know that 5 > 2.
∴ d(V, B) = 5 − 2 = 3
The co-ordinates of points U and L are −5 and −3 respectively. We know that −3 > −5.
∴ d(U, L) = −3 − (−5) = −3 + 5 = 2

#### Question 1:

Write the following statements in ‘if-then’ form.

(i) The opposite angles of a parallelogram are congruent.
(ii) The diagonals of a rectangle are congruent.
(iii) In an isosceles triangle, the segment joining the vertex and the mid point of the base is perpendicular to the base.

(i) If a quadrilateral is a parallelogram, then the opposite angles of that quadrilateral are congruent.

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

(iii) If a triangle is an isosceles, then the segment joining the vertex and the mid point 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.
(ii) If a pair of the interior angles made by a transversal of two lines are supplementary then the lines are parallel.
(iii) The diagonals of a rectangle are congruent.

(i) If the alternate angles made by the transversal with the two lines are congruent, then the lines are parallel.

(ii) If the two parallel lines are intersected by a transversal, then the pair of interior angles are supplementary.

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

#### 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

(ii) How many points are there in the intersection of two distinct lines ?
(A) infinite (B) two (C) one (D) not a single

(iii) How many lines are determined by three distinct points ?
(A) two (B) three (C) one or three (D) six

(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

(v) If P - Q - R and d(P,Q) = 2, d(P,R) = 10, then find d(Q,R).
(A) 12   (B) 8    (C) $\sqrt{96}$   (D) 20

(i) Every segment has one and only one midpoint.
Hence, the correct answer is option (A).

(ii) It is known that, two distinct lines intersect at one point.
Hence, the correct answer is option (C).

(iii) Consider the 3 distinct points as P, Q and R.
Suppose the points P, Q and R are collinear.

So, only one line is determined by the points P, Q and R.
Suppose the points P,Q and R are non collinear.

So, three lines can be determined by the points P, Q and R.
Hence, the correct answer is option(C).

(iv) The co-ordinates of points A and B are $-$2 and 5 respectively. We know that 5 > ​$-$2.
d(A, B) = 5 − (−2) = 5 + 2 = 7
Hence, the correct answer is option(C).

(v) It is given that, point Q is between point P and point R.

We have, d(P,Q) = 2;  d(P,R) = 10
Now, d(P,R) = d(P,Q) + d(Q,R)
∴ d(Q,R) = d(P,R) − d(P,Q) = 10 − 2 = 8
Hence, the correct answer is option (B).

#### 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)

The co-ordinates of points P and Q are 3 and −5 respectively. We know that 3 > −5.
Now, d(P, Q) = 3 − (−5) = 3 + 5 = 8
The co-ordinates of points Q and R are −5 and 6 respectively. We know that 6 > −5.
Now, d(Q, R) = 6 − (−5) = 6 + 5 = 11
The co-ordinates of points P and R are 3 and 6 respectively. We know that 6 > 3.
Now, d(P, R) = 6 − 3 = 3

(i) d(P, Q) + d(Q, R) = 8 + 11 = 19; d(P, R) = 3
So, d(P, Q) + d(Q, R) ≠ d(P, R)
Hence, the given statement is false.

(ii) d(P, R) + d(R, Q) = d(P, R) + d(Q, R) = 3 + 11 = 14; d(P, Q) = 8
So, d(P, R) + d(R, Q) ≠ d(P, Q)
Hence, the given statement is false.

(iii) d(R,P) + d(P,Q) = d(P, R) + d(P,Q) = 3 + 8 = 11; d(R,Q) = d(Q, R) = 11
So, d(R,P) + d(P,Q) = d(R,Q)
Hence, the given statement is true.

(iv) d(P,Q) $-$ d(P,R) = 8 − 3 = 5; d(Q,R) = 11
So, d(P,Q) $-$ d(P,R) ≠ d(Q,R)
Hence, the given statement is false.

#### Question 3:

Co-ordinates of some pairs of points are given below. Hence find the distance between each pair.

(i) 3, 6
(ii) − 9, $-$ 1
(iii)$-$ 4, 5
(iv) x,$-$ 2
(v) x + 3, x$-$ 3
(vi) $-$25,$-$47
(vii) 80, $-$ 85

(i) Let the co-ordinates of A and B are 3 and 6 respectively. We know that 6 > 3
d(A, B) = 6 − 3 = 3

(ii) Let the co-ordinates of C and D are −9 and −1 respectively. We know that −1 > −9
d(C, D) = −1 − (−9) = −1 + 9 = 8

(iii) Let the co-ordinates of E and F are −4 and 5 respectively. We know that 5 > −4
d(E, F) = 5 − (−4) = 5 + 4 = 9

(iv) Let the co-ordinates of P and Q are x and −2 respectively. Suppose x > 0, then x >  −2.
d(P, Q) = x − (−2) = x + 2

(v) Let the co-ordinates of R and S are x + 3 and x − 3 respectively. Suppose > 0, then x + 3 > x − 3
d(R, S) = (x + 3) − (x − 3) = x + 3 − x + 3 = 2x

(vi) Let the co-ordinates of L and M are −25 and −47 respectively. We know that −25 > −47
d(L, M) = −25 − (−47) = −25 + 47 = 22

(vii) Let the co-ordinates of G and H are 80 and −85 respectively. We know that 80 > −85
d(G, H) = 80 − (−85) =80 + 85 = 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.

The co-ordinates of point P on the number line is −7. Now, there will be two points, one on the left of point P and the other on the right of point P on the number line which are at a distance of 8 units from point P.
Let the point R is on the right of point P and point Q is on the left of point P each at a distance of 8 units from point P.
The co-ordinate of point R will be larger and co-ordinate of point Q will be smaller in comparison to the co-ordinate of point P.
Now, d(P, R) = 8
So, co-ordinate of R − co-ordinate of P = 8
∴ co-ordinate of R = 8 + co-ordinate of P = 8 + (−7) = 8 − 7 = 1
Also, d(Q, P) = 8
So, co-ordinate of P − co-ordinate of Q = 8
∴ co-ordinate of Q =  co-ordinate of P − 8 = −7 − 8 = −15

Hence, the co-ordinates of the required points on the number line which are at a distance of 8 units from the point P are 1 and −15.

#### Question 5:

(i) If A - B - C and d(A,C) = 17, d(B,C) = 6.5 then d(A,B) = ?
(ii) If P - Q - R and d(P,Q) = 3.4, d(Q,R)= 5.7 then d(P,R) = ?

(i)

We have, d(A, C) = 17; d(B, C) = 6.5
Now, d(A, C) = d(A, B) + d(B, C)
So, d(A, B) = d(A, C) − d(B, C) = 17 − 6.5
∴ d(A, B) = 10.5

(ii)

We have, d(P, Q) = 3.4; d(Q, R) = 5.7
Now, d(P, R) = d(P, Q) + d(Q, R) = 3.4 + 5.7
∴ 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 ?

The co-ordinates of point A on the number line is 1. Now, there will be two points, one on the left of point A and the other on the right of point A on the number line which are at a distance of 7 units from point A.
Let the point C is on the right of point A and point B is on the left of point A each at a distance of 7 units from point A.
The co-ordinate of point C will be larger and co-ordinate of point B will be smaller in comparison to the co-ordinate of point A.
Now, d(A, C) = 7
So, co-ordinate of C − co-ordinate of A = 7
∴ co-ordinate of C = 7 + co-ordinate of A = 7 + 1= 8
Also, d(B, A) = 7
So, co-ordinate of A − co-ordinate of B = 7
∴ co-ordinate of B =  co-ordinate of A − 7 = 1 − 7 = −6

Hence, the co-ordinates of the required points on the number line which are at a distance of 7 units from the point A are 8 and −6.

#### Question 7:

Write the following statements in conditional form.
(i) Every rhombus is a square.
(ii) Angles in a linear pair are supplementary.
(iii) A triangle is a figure formed by three segments.
(iv) A number having only two divisors is called a prime number.

(i) If the given quadrilateral is a square, then it must be a rhombus.
(ii) If the given two angles are forming a linear pair, then they are supplementary.
(iii) If the given figure is a triangle, then it is formed by three segments.
(iv) If the given number is having only two divisors, then it is 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.
(ii) If the sum of measures of two angles is 900 then they are complement of each other.
(iii) If the corresponding angles formed by a transversal of two lines are congruent then the two lines are parallel.
(iv) If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.

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

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

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

(iv) If the given number is divisible by 3, then the sum of the digits of the number is 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.
(ii) The diagonals of a parallelogram bisect each other.

(i) Antecedent : All the sides of a triangle are congruent.
Consequent : Its all angles are congruent.

(ii) The statement can be written in conditional form as, 'If the given quadrilateral is a parallelogram, then its diagonals bisect each other.
Antecedent : The given quadrilateral is a parallelogram.
Consequent : Its 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.
(ii) If angles in a linear pair are congruent then each of them is a right angle.
(iii) If the altitudes drawn on two sides of a triangle are congruent then those two sides are congruent.

(i) The given statement can be written in conditional form as, ' If the given two traingles are equilateral, then they are similar.'
Antecedent : The given two triangles are equilateral.
Consequent : They are similar.

Here, ∆ABC and ∆PQR are equilateral triangles, so they are similar to each other.

(ii) Antecedent : The angles in a linear pair are congruent.
Consequent : Each of them is a right angle.

Here, ∠AOC and ∠BOC forming a linear pair are congruent to each other, so each of them is a right angle.

(iii) Antecedent : The altitudes drawn on two sides of a triangle are congruent.
Consequent : Those two sides are congruent.

Here, BL and CM are the altitudes drawn on two sides AC and AB respectively of ∆ABC and are congruent, so side AB is congruent to side AC.

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