Step: 1

[∠ SQP is a straight angle.]

Step: 2

[∠ TRP is a straight angle.]

Step: 3

180^{o} - (∠ PQR) > 180^{o} - (∠ PRQ)

[∠ SQR > ∠ TRQ.]

Step: 4

[Simplify.]

Step: 5

PQ > PR

[In a triangle the side opposite to the greater angle is greater.]

Correct Answer is : PQ > PR

Step: 1

[Triangle Inequality Theorem.]

Step: 2

[In a triangle the difference of any two sides is less than the third side.]

Step: 3

Step: 4

Step: 5

2 < z < 12

[Step 2 and step 4.]

Correct Answer is : 2 < z < 12

Step: 1

BC < 14 + 9

[Triangle Inequality Theorem.]

Step: 2

BC < 23 cm

Step: 3

BC > 14 - 9

[In a triangle, the difference of the lengths of any two sides is less than the third side.]

Step: 4

BC > 5 cm

Step: 5

5 cm < BC < 23 cm

Step: 6

As 9 lies between 5 and 23, BC can be 9 cm.

Correct Answer is : 9 cm

Step: 1

BC < 17 + 6

[Triangle Inequality Theorem.]

Step: 2

BC < 23 cm

Step: 3

BC > 17 - 6

[In a triangle, the difference of the lengths of any two sides is less than the third side.]

Step: 4

BC > 11 cm

Step: 5

11 cm < BC < 23 cm

Step: 6

From the values given in the options, 14 cm lies between 11 cm and 23 cm.

Step: 7

So, the length of BC is 14 cm.

Correct Answer is : 14 cm

Step: 1

AB > BC > CA ⇒ ∠ C > ∠ A > ∠ B

[In a triangle, the greater angle is opposite to the greater side.]

Step: 2

Option (C) satisfies the above condition as for m ∠ A = 40, m ∠ B = 30 and m ∠ C = 110, ∠ C > ∠ A > ∠ B.

Correct Answer is : 40, 30, 110

Step: 1

Smallest angle + medium angle + largest angle = 180^{o}

[Triangle-Angle-Sum theorem.]

Step: 2

40^{o} + (Medium angle) + (Largest angle) = 180^{o}

[Substitute.]

Step: 3

Medium angle = 140 - (largest angle)

[Simplify.]

Step: 4

Medium angle > smallest angle

Step: 5

140 - (largest angle) > 40^{o}

[Step 3.]

Step: 6

Largest angle < 100^{o}

[Simplify.]

Correct Answer is : less than 100^{o}

Step: 1

Step: 2

In ΔADB, AB > AD

[In a right triangle, hypotenuse is the longest side.]

Step: 3

Similarly, BC > BE and CA > CF

Step: 4

AB + BC + CA > AD + BE + CF

[Add the three inequalities above.]

Step: 5

Correct Answer is : p > h

Step: 1

Let a , b , c be the length of the sides and m _{1}, m _{2}, m _{3} be the length of the medians.

Step: 2

[Produce AD to E such that AD = DE and join C to E.]

Step: 3

ΔBDA ≅ ΔCDE

[SAS postulate.]

Step: 4

CE = BA = c

[Step 3.]

Step: 5

In ΔAEC, AC + CE > AE

[Triangle Inequality Theorem.]

Step: 6

[Substitute.]

Step: 7

Similarly, for other medians, we get c + a > 2m _{2} and a + b > 2m _{3}

Step: 8

(b + c ) + (c + a ) + (a + b ) > 2m _{1} + 2m _{2} + 2m _{3}

[Add the three inequalities.]

Step: 9

2(a + b + c ) > 2(m _{1} + m _{2} + m _{3})

[Add.]

Step: 10

(a + b + c ) > m _{1} + m _{2} + m _{3}

[Simplify.]

Step: 11

Correct Answer is : p > r

Step: 1

In ΔA B C , ∠ C = 180^{o} - (∠ A + ∠ B )

[Triangle Angle Sum theorem.]

Step: 2

[Substitute.]

Step: 3

[Substitute.]

Step: 4

Step: 5

[In a triangle, the greatest side is opposite to the greatest angle.]

Correct Answer is : BC

Step: 1

An obtuse angle is the greatest angle in the triangle.

[Triangle-Angle-Sum theorem.]

Step: 2

Side opposite to the greatest angle i.e., the obtuse angle is the greatest.

Correct Answer is : the greatest

Step: 1

[As PQ = PR .]

Step: 2

[Triangle Angle Sum theorem.]

Step: 3

[Substitute.]

Step: 4

[Simplify.]

Step: 5

Step: 6

[In a triangle, the side opposite to the greater angle is greater.]

Step: 7

Correct Answer is : Q R < P R

Step: 1

[Triangle-Angle-Sum theorem.]

Step: 2

[Substitute.]

Step: 3

[Simplify.]

Step: 4

Step: 5

[In a triangle, the greater side is opposite to the greater angle.]

Correct Answer is : A C > A B

Step: 1

Step: 2

In ΔXOY, ∠ OYX > ∠ OXY ⇒ OX > OY

[In a triangle the greater side is opposite to the greater angle.]

Step: 3

OY < OX.

Correct Answer is : OY < OX

Step: 1

2x > 13

[Sum of two sides shall be greater than the third side.]

Step: 2

Minimum value of x shall be 7 units.

[x is to be a whole number.]

Correct Answer is : 7 units

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