# RD SHARMA Solutions for Class 9 Maths Chapter 11 - Triangle and its Angles

## Chapter 11 - Triangle and its Angles Exercise Ex. 11.1

Two angles of a triangle are equal and the third angle is greater than each of those angles by 30^{o}. Determine all the angles of the triangle.

Can a triangle have:

(i) Two right angles?

(ii) Two obtuse angles?

(iii) Two acute angles?

(iv) All angles more than 60^{o}?

(v) All angles less than 60^{o}?

(vi) All angles equal to 60^{o}?

Justify your answer in each case.

(i) No

As two right angles would sum up to 180^{o}, and we know that the sum of all three angles of a triangle is 180^{o}, so the third angle will become zero. This is not possible, so a triangle cannot have two right angles.

(ii) No

A triangle cannot have 2 obtuse angles, since then the sum of those two angles will be greater than 180^{o }which is not possible as the sum of all three angles of a triangle is 180^{o}.

(iii) Yes

A triangle can have 2 acute angles.

(iv) No

The sum of all the internal angles of a triangle is 180^{o}. Having all angles more than 60^{o} will make that sum more than 180^{o}, which is impossible.

(v) No

The sum of all the internal angles of a triangle is 180^{o}. Having all angles less than 60^{o} will make that sum less than 180^{o}, which is impossible.

(vi) Yes

The sum of all the internal angles of a triangle is 180^{o}. So, a triangle can have all angles as 60^{o}. Such triangles are called equilateral triangles.

## Chapter 11 - Triangle and its Angles Exercise Ex. 11.2

The exterior angles, obtained on producing both the base of a triangle both ways are 104^{o} and 136^{o}. Find all the angles of the triangle.

In fig., the sides BC, CA and AB of a ΔABC have been produced to D, E, and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the ΔABC.

In fig., AC ⊥ CE and ∠A : ∠B : ∠C = 3 : 2 : 1, find the value of ∠ECD.

In fig. AB ∥ DE. Find ∠ACD.

Which of the following statements are true (T) and which are false (F):

Fill in the blanks to make the following statements true:

(i) Sum of the angle of triangle is ______ .

(ii) An exterior angle of a triangle is equal to the two ______ opposite angles.

(iii) An exterior angle of a traingle is always _______ than either of the interior oppsite angles.

(iv) A traingle cannot have more than ______ right angles.

(v) A triangles cannot have more than ______ obtuse angles.

(i) 180^{o}

(ii) interior

(iii) greater

(iv) one

(v) one

In fig., AB divides ∠DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.

In fig., AM ⊥ BC and AN is the bisector of ∠A. If ∠B = 65° and ∠C = 33°, find ∠MAN.

In fig. AE bisects ∠CAD and ∠B = ∠C. Prove that AE ∥ BC.

## Chapter 11 - Triangle and its Angles Exercise 11.25

If all the three angles of a triangle are equal, then each one of them is equal to

(a) 90°

(b) 45°

(c) 60°

(d) 30°

Let the measure of each angle be x°.

Now, the sum of all angles of any triangle is 180°.

Thus, x° + x° + x° = 180°

i.e. 3x° = 180°

i.e. x° = 60°

Hence, correct option is (c).

If two acute angles of a right triangle are equal, then each acute is equal to

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Let the measure of each acute angle of a triangle be x°.

Then, we have

x° + x° + 90° = 180°

i.e. 2x° = 90°

i.e. x° = 45°

Hence, correct option is (b).

An exterior angle of a triangle is equal to 100° and two interior opposite angle are equal. Each of these angles is equal to

(a) 75°

(b) 80°

(c) 40°

(d) 50°

If one angle of a triangle is equal to the sum of the other two angles, then the triangle is

(a) an isosceles triangle

(b) an obtuse triangle

(c) an equilateral triangle

(d) a right triangle

Let the three angles of a triangle be A, B and C.

Now, A + B + C = 180°

If A = B + C

Then A + (A) = 180°

i.e. 2A = 180°

i.e. A = 90°

Since, one of the angle is 90°, the triangle is a Right triangle.

Hence, correct option is (d).

In a triangle, an exterior angle at a vertex is 95° and its one of the interior opposite angle is 55°, then the measure of the other interior angle is

(a) 55°

(b) 85°

(c) 40°

(d) 9.0°

If the sides of a triangle are produced in order, then the sum of the three exterior angles so formed is

(a) 90°

(b) 180°

(c) 270°

(d) 360°

An exterior angle of a triangle is 108° and its interior opposite angles are in the ratio 4 : 5. The angles of the triangle are

(a) 48°, 60°, 72°

(b) 50°, 60°, 70°

(c) 52°, 56°, 72°

(d) 42°, 60°, 76°

## Chapter 11 - Triangle and its Angles Exercise 11.26

In figure, x + y =

(a) 270°

(b) 230°

(c) 210°

(d) 190°

If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?

(a) 25°

(b) 30°

(c) 45°

(d) 60°

## Chapter 11 - Triangle and its Angles Exercise 11.27

In figure, what is z in terms of x and y?

(a) x +y + 180°

(b) x + y - 180°

(c) 180° - (x + y)

(d) x + y + 360°

In figure, what is the value of x?

(a) 35

(b) 45

(c) 50

(d) 60

## Chapter 11 - Triangle and its Angles Exercise 11.28

In figure, the value of x is

(a) 65°

(b) 80°

(c) 95°

(d) 120°

## Chapter 11 - Triangle and its Angles Exercise 11.29

If the bisectors of the acute angles of a right triangle meet at O, then the angle at O between the two bisectors is

(a) 45°

(b) 95°

(c) 135°

(d) 90°

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