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**Q1. In each of the following, there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle:**

**(i) 5, 7, 9**

**(ii) 2, 10.15**

**(iii) 3, 4, 5**

**(iv) 2, 5, 7**

**(v) 5, 8, 20**

**Solution. **

(i) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side. Here, 5+7>9, 5+9>7, 9+7>5

(ii) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.

(iii) Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of triangle is always greater than the third side. Here, 3+4 >5, 3+5> 4, 4+5> 3

(iv) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case. Here, 2 + 5 = 7

(v) No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case. Here, 5 + 8 <20

**Q2. In Fig, P is the point on the side BC. Complete each of the following statements using symbol â€˜ =â€™,â€™ > â€˜or â€˜ < â€˜so as to make it true:**

**(i) APâ€¦ AB+ BP**

**(ii) APâ€¦ AC + PC**

**Solution. **

(i) In triangle APB, AP < AB + BP because the sum of any two sides of a triangle is greater than the third side.

(ii) In triangle APC, AP < AC + PC because the sum of any two sides of a triangle is greater than the third side.

In triangles ABP and ACP, we can see that:

AP < AB + BPâ€¦(i) (Because the sum of any two sides of a triangle is greater than the third side)

AP < AC + PCâ€¦(ii) (Because the sum of any two sides of a triangle is greater than the third side)

On adding (i) and (ii), we have:

AP + AP < AB + BP + AC + PC

2AP < AB + AC + BC (BC = BP + PC)

AP < (AB-FAC+BC)

**Q3. P is a point in the interior of Î”ABC as shown in Fig. State which of the following statements are true (T) or false (F):**

**(i) AP+ PB< AB**

**(ii) AP+ PC> AC**

**(iii) BP+ PC = BC**

**Solution. **

(i) False

We know that the sum of any two sides of a triangle is greater than the third side: it is not true for the given triangle.

(ii) True

We know that the sum of any two sides of a triangle is greater than the third side: it is true for the given triangle.

(iii) False

We know that the sum of any two sides of a triangle is greater than the third side: it is not true for the given triangle.

**Q4. O is a point in the exterior of Î”ABC. What symbol â€˜>â€™,â€™<â€™ or â€˜=â€™ will you see to complete the statement OA+OBâ€¦.AB? Write two other similar statements and show that**

**Solution. **

Because the sum of any two sides of a triangle is always greater than the third side, in triangle OAB, we have:

OA+OB> AB â€”(i)

OB+OC>BC â€”-(ii)

OA+OC > CA â€”â€“(iii)

On adding equations (i), (ii) and (iii) we get :

OA+OB+OB+OC+OA+OC> AB+BC+CA

2(OA+OB+OC) > AB+BC +CA

**Q5. In â–³ABC, âˆ B=30 ^{âˆ˜}, âˆ C=50^{âˆ˜}. Name the smallest and the largest sides of the triangle.**

**Solution. **

Because the smallest side is always opposite to the smallest angle, which in this case is 30^{âˆ˜}, it is AC. Also, because the largest side is always opposite to the largest angle, which in this case is 100^{âˆ˜},, it is BC.