#### Chapter 9 Triangle and its Angles R.D. Sharma Solutions for Class 9th Exercise 9.1

**Exercise 9.1**

1.Â In a Î” ABC, if âˆ A = 55Â°, âˆ B = 40Â°, find âˆ C.

**Solution**

2. If the angles of a triangle are in the ratio 1:2:3, determine three angles.

**Solution**

Let the angles of the given triangle be of xÂ°, 2xÂ° and 3xÂ°. Then,

âˆ´ x+2x+3x = 180Â° (The sum of three angles of a triangle is 180Â°)

â‡’ 6x = 180Â°

â‡’ x = 30Â°

3. The angles of a triangle are (x-40)Â°, (x-20 and (1/2x -10)Â°. Find the value of x.

**SolutionÂ**

4. The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10Â°, find the three angles.

**Solution**

Let the angles of a triangle xÂ°, (x+10)Â° and (x+20)Â°.

Since, the difference between two consecutive angles is 10Â°.

âˆ´ x+(x+10)+(x+20) = 180Â (Sum of the three angles of a triangle is 180Â°.

â‡’ 3x+30 = 180

â‡’ 3x = 150

â‡’ x = 50

Therefore, the angles of the given triangle are 50Â°, (50+10)Â° and (50+20)Â° i.e. 50Â°, 60Â° and 70Â°.

5. Two angles of a triangle are equal and the third angle is greater than each of those angles by 30Â°. Determine all the angles of the triangle.

**Solution**

Let the two equal angles are xÂ°, then the third angle will be (x+30)Â°.

âˆ´ x+x+(x+30) = 180Â (Sum of the three angles of a triangle is 180Â°.)

â‡’ 3x+30 = 180

â‡’ 3x = 150

â‡’ x = 50

Therefore, the angles of the given triangle are 50Â°, 50Â° and 80Â°.

6. If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.

**Solution**

Let ABC be a triangle such that,

âˆ A = âˆ B + âˆ CÂ (Since, one angle is sum of the other two angles.)

âˆ´ âˆ A + (âˆ B +âˆ C) = 180Â (Sum of the three angles of a triangle is 180Â°.)

â‡’ âˆ A + âˆ A = 180

â‡’ 2âˆ A = 180

â‡’ âˆ A = 90

Hence, the given triangle is a right angled triangle.

7. ABC is a triangle in which âˆ A = 72Â°, the internal bisectors of angles B and C meet in O. Find the magnitude of âˆ BOC.

**Solution**

8. The bisectors of base angles of a triangle cannot enclose a right angle in any case.

**Solution**

Let ABC be a triangle and BO and CO be the bisectors of the base angle âˆ ABC and âˆ ACB respectively.

9. If the bisectors of the base angles of a triangle enclose an angle of 135Â°, prove that the triangle is a right triangle.

**Solution**

10. In a Î” ABC, âˆ ABC = âˆ ACB and the bisectors of âˆ ABC and âˆ ACB intersect at O such that âˆ BOC = 120Â°. Show that âˆ A =âˆ B =âˆ C = 60Â°.

**Solution**

11. Can a triangle have:

(i) Two right angles?

(ii) Two obtuse angles?

(iii) Two acute angles?

(iv) All angles more than 60Â°?

(v) All angles less than 60Â°?

(vi) All angles equal to 60Â°?

**Solution**

(i) Let a triangle ABC has two angles âˆ B and âˆ C equal to 90. We know that sum of the three angles of a triangle is 180Â°.

âˆ A+âˆ B+âˆ C = 180Â°

â‡’ 90Â° + 90Â° + âˆ C = 180Â°Â [âˆ A = 90Â° and âˆ B = 90Â°]

â‡’ 180 + âˆ C = 180

â‡’ âˆ C = 0Â°

Hence, if two angles are equal to 90Â°, then the third one will be equal to zero which implies that A, B, C is collinear, or we can say ABC is not a triangle A triangle canâ€™t have two right angles.

(ii) Let a triangle ABC has two obtuse anglesÂ âˆ B and âˆ C.

This implies that sum of only two angles will be equal to more than 180Â° which contradicts the theorem sum of all angles in a triangle is always equals 180Â°.

Therefore, a triangle canâ€™t have two obtuse angles.

(iii) Let a triangle ABC has two acute angles âˆ B and âˆ C.

This implies that sum of two angles will be less than 180Â°. Hence third angle will be the difference of 180Â° and sum of both acute angles

Therefore, a triangle can have two acute angles.

(iv) Let a triangle ABC having angles âˆ A, âˆ B and âˆ C are more than 60Â°.

This implies that the sum of three angles will be more than 180Â° which contradicts the theorem sum of all angles in a triangle is always equals 180Â°.

Therefore, a triangle canâ€™t have all angles more than 60Â°.

(v) Let a triangle ABC having angles âˆ A, âˆ B and âˆ C are less than 60Â°.

This implies that the sum of three angles will be less than 180Â° which contradicts the theorem sum of all angles in a triangle is always equals 180Â°.

Therefore, a triangle canâ€™t have all angles less than 60Â°.

(vi) Let a triangle ABC having angles âˆ A, âˆ B and âˆ C all equal to 60Â°.

This implies that the sum of three angles will be equal to 180Â° which satisfies the theorem sum of all angles in a triangle is always equals 180Â°.

Therefore, a triangle can have all angles equal to 60Â°.

12. If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.

**Solution**