myhelper: RD Sharma 2020 solution class 9 chapter 11 Triangles and Its Angles FBQS

RD Sharma 2020 solution class 9 chapter 11 Triangles and Its Angles FBQS

FBQS

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Question 1:

The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is a/an ___________ triangle.

Answer 1:

Let the measure of three angles of the triangle be 5x, 3x and 7x.

Now,

$5 x + 3 x + 7 x = 180 °$ (Angle sum property of triangle)

$\Rightarrow 15 x = 180 °$

$\Rightarrow x = \frac{180 °}{15} = 12 °$

$∴ 5 x = 5 \times 12 ° = 60 °, 3 x = 3 \times 12 ° = 36 °$ and $7 x = 7 \times 12 ° = 84 °$

So, the measure of the angles of the triangle are 60º, 36º and 84º.

A triangle, each of whose angles is acute, is called an acute triangle. Thus, the triangle is an acute triangle.

The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is a/an __acute__ triangle.

Question 2:

Angles of a triangle are in the ratio 2 : 4 : 3. The measure of the smallest angle of the triangle is ________.

Answer 2:

Let the measure of three angles of the triangle be 2x, 4x and 3x.

Now,

$2 x + 4 x + 3 x = 180 °$ (Angle sum property of triangle)

$\Rightarrow 9 x = 180 °$

$\Rightarrow x = \frac{180 °}{9} = 20 °$

$∴ 2 x = 2 \times 20 ° = 40 °, 4 x = 4 \times 20 ° = 80 °$ and $3 x = 3 \times 20 ° = 60 °$

So, the measure of the angles of the triangle are 40º, 80º and 60º.

Thus, the measure of the smallest angle of the triangle is 40º.

Angles of a triangle are in the ratio 2 : 4 : 3. The measure of the smallest angle of the triangle is ___40º___.

Question 3:

The number of triangles that can be drawn the measure of whose angles are 53°, 64° and 63°, is _________.

Answer 3:

We know that the sum of the angles of a triangle is 180º.

The given angles are 53°, 64° and 63°.

Sum of the given angles = 53° + 64° + 63° = 180º

Here, the sum of the angles of the triangle is 180º. But, the measure of sides of the triangles is not known. So, infinitely many triangles can be drawn and sum of the angles of every triangle is 180º.

The number of triangles that can be drawn the measure of whose angles are 53°, 64° and 63°, is __infinite many__.

Question 4:

If the measure of one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles is _________.

Answer 4:

In ∆ABC, ∠A = 130º.

Also, OB and OC are the bisectors of ∠B and ∠C, respectively.

$∴ ∠ OBC = \frac{∠ B}{2} . . . . . (1)$

Similarly, $∠ OCB = \frac{∠ C}{2} . . . . . (2)$

Now,

$∠ A + ∠ B + ∠ C = 180 °$ (Angle sum property of triangle)

$\Rightarrow 130 ° + ∠ B + ∠ C = 180 °$

$\Rightarrow ∠ B + ∠ C = 180 ° - 130 ° = 50 °$ .....(3)

In ∆BOC,

∠OBC + ∠OCB + ∠BOC = 180º (Angle sum property of triangle)

$\Rightarrow \frac{∠ B}{2} + \frac{∠ C}{2} + ∠ BOC = 180 ° [Using (1) and (2)] \Rightarrow \frac{∠ B + ∠ C}{2} + ∠ BOC = 180 ° \Rightarrow \frac{50 °}{2} + ∠ BOC = 180 ° [Using (3)] \Rightarrow ∠ BOC = 180 ° - 25 ° = 155 °$

If the measure of one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles is ___155º___.

Question 5:

An exterior angle of a triangle is 105° and its two interior opposite angles are equal. The measure of each of these two angles is __________.

Answer 5:

Let the measure of each of the two interior opposite angles be x.

Measure of exterior angle of triangle = 105°

We know that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

$∴ x + x = 105 °$

$\Rightarrow 2 x = 105 °$

$\Rightarrow x = \frac{105 °}{2} = 52.5 °$

An exterior angle of a triangle is 105° and its two interior opposite angles are equal. The measure of each of these two angles is ___52.5º___.

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Question 6:

The number of triangles that can be drawn with the measure of each angle less than 60°, is __________.

Answer 6:

Let the three angles of the triangle be x, y and z.

It is given that,

x < 60º, y < 60º and z < 60º

Now,

x + y + z < 60º + 60º + 60º

⇒ x + y + z < 180º

Or Sum of the angles of the triangle < 180º

This is not possible as the sum of the angles of a triangle is always equal to 180º.

Thus, no triangle can be drawn with the measure of each angle less than 60°.

The number of triangles that can be drawn with the measure of each angle less than 60°, is ____0____.

Question 7:

A triangle cannot have two ___________ angles.

Answer 7:

An angle whose measure is more than 90º but less than 180º is called an obtuse angle.

If two angles of a triangle are obtuse, then the sum of these two obtuse angles would be more than 180º.

So, the sum of the three angles of the triangle in this case would be more than 180º. This is not possible as the sum of three angles of a triangle is always equal to 180º.

Thus, a triangle cannot have two obtuse angles.

A triangle cannot have two ___obtuse ___ angles.

Question 8:

All angles of a triangle can be __________ angles.

Answer 8:

An angle whose measure is less than 90º is called an acute angle. A triangle can have three acute angles such that their sum is 180º. A triangle which has all acute angles is called an acute triangle.

All angles of a triangle can be __acute__ angles.

Question 9:

In a ∆ABC, if ∠A < ∠B < 45°, then ∆ABC is a/an ________ triangle.

Answer 9:

In ∆ABC, ∠A < ∠B < 45°. This means that the measure of ∠A is less than 45º and measure of ∠B is less than 45º. Also, the measure of ∠A is less than measure of ∠B.

So, ∠A < 45° and ∠B < 45°

∴ ∠A + ∠B < 45° + 45°

⇒ ∠A + ∠B < 90°

Adding ∠C to both sides, we have

∠A + ∠B + ∠C < 90° + ∠C .....(1)

We know

∠A + ∠B + ∠C = 180° (Angle sum property of triangle)

So,

180° < 90° + ∠C [Using (1)]

Or 90° + ∠C > 180°

⇒ ∠C > 180° − 90°

⇒ ∠C > 90°

A triangle with one angle an obtuse angle is known as an obtuse triangle. So, ∆ABC is an obtuse triangle.

In a ∆ABC, if ∠A < ∠B < 45°, then ∆ABC is a/an __obtuse__ triangle.

Question 10:

In a triangle ABC, if ∠A > ∠B > ∠C and the measures of ∠A, ∠B and ∠C in degrees are integers, then the least possible values of A, B and C are _______ and ______ respectively.

Answer 10:

It is given that, in ∆ABC, ∠A > ∠B > ∠C and the measures of ∠A, ∠B and ∠C in degrees are integers.

So, the least value of ∠C is 1º and the least value of ∠B is 2º.

In ∆ABC,

∠A + ∠B + ∠C = 180° (Angle sum property of triangle)

∴ ∠A + 2° + 1° = 180°

⇒ ∠A = 180° − 3° = 177°

In a triangle ABC, if ∠A > ∠B > ∠C and the measures of ∠A, ∠B and ∠C in degrees are integers, then the least possible values of A, B and C are __177°, 2°__ and __1°_ respectively.

Note: The value of ∠A depends upon the values of ∠B and ∠C. The least value of ∠A would be 61º. But, it that case the values of ∠B or ∠C would not be least.

Question 11:

The measures of three angles of a triangle are in the ratio 1 : 2 : 3. Then, the triangle is a __________ triangle.

Answer 11:

Let the measure of three angles of the triangle be x, 2x and 3x.

Now,

$x + 2 x + 3 x = 180 °$ (Angle sum property of triangle)

$\Rightarrow 6 x = 180 °$

$\Rightarrow x = \frac{180 °}{6} = 30 °$

$∴ 2 x = 2 \times 30 ° = 60 °$ and $3 x = 3 \times 30 ° = 90 °$

So, the measure of the angles of the triangle are 30º, 60º and 90º.

Now, a triangle with one angle a right angle is called a right triangle. Thus, the triangle is a right triangle.

The measures of three angles of a triangle are in the ratio 1 : 2 : 3. Then, the triangle is a ___right___ triangle.

Question 12:

The internal bisectors of ∠B and ∠C of ∆ABC meet at O. If B + C = 100°, then ∠BOC = __________.

Answer 12:

In ∆ABC, ∠B + ∠C = 100°.

Also, OB and OC are the bisectors of ∠B and ∠C, respectively.

$∴ ∠ OBC = \frac{∠ B}{2} . . . . . (1)$

Similarly, $∠ OCB = \frac{∠ C}{2} . . . . . (2)$

In ∆BOC,

∠OBC + ∠OCB + ∠BOC = 180º (Angle sum property of triangle)

$\Rightarrow \frac{∠ B}{2} + \frac{∠ C}{2} + ∠ BOC = 180 ° [Using (1) and (2)] \Rightarrow \frac{∠ B + ∠ C}{2} + ∠ BOC = 180 ° \Rightarrow \frac{100 °}{2} + ∠ BOC = 180 ° (Given) \Rightarrow ∠ BOC = 180 ° - 50 ° = 130 °$

The internal bisectors of ∠B and ∠C of ∆ABC meet at O. If B + C = 100°, then ∠BOC = ____130º____.