## Karnataka State Syllabus Class 8 Maths Chapter 6 Theorems on Triangles Additional Questions

Question 1.

Fill up the blanks to make the following statements true

a. Sum of the angles of a triangle in 180°

b. An exterior angle of a triangle is equal to the sum of interior opposite angles.

c. An exteior angle of a triangle is always more than either of the interior opposite angles.

d. A triangle can not have more than one right angles.

e. A triangle can not have more than one obtuse angle.

Question 2.

Choose the correct answer from the given alternatives.

a. In a triangle ABC, ∠A = 80° and AB = BAC then ∠B is __________

A. 50°

B. 60°

C. 40°

D. 70°

Solution:

A. 50°

b. In right-angled triangle, ∠A is right angle and ∠B = 35° then ∠C is _______

A. 65°

B. 55°

C. 75°

D. 45°

Solution:

B. 55°

c. In ∆ABC, ∠B = ∠C = 45°, then the triangle is ________

A. Right-angled

B. Acute angled

C. Obtuse angled

D. Equilateral triangle

Solution:

A. Right-angled

d. In an equilateral triangle, each exterior angle is ________

A. 60°

B. 90°

C. 120°

D. 150°

Solution:

C. 120°

e. Sum of the three exterior angles of a triangle is __________

A. two right angles

B. three right angles

C. one right angle

D. four right angles

Solution:

D. Four right angles

Question 3.

In a triangle ABC, ∠B = 70° find ∠A + ∠C

Solution:

∠A + ∠B + ∠C = 180° [Sum of the angles of a triangle is 180°]

∠A + 70 + ∠C = 180°

∠A + ∠C = 180° – 70

∠A + ∠C = 110°

Question 4.

In a triangle ABC, ∠A =110° and AB = AC find ∠B and ∠C

Solution:

AB = AC

∴ ∠C = ∠B [Base angles of an isosceles triangle]

∠A + ∠B + ∠C = 180° [Sum of the angles of a triangle is 180°]

110° + ∠B + ∠B = 180°

2∠B = 180 – 110

2∠B = 70°

∠B = 702

∠B = 35°

∠B = ∠C = 35°

Question 5.

If three angles of a triangle are in the ratio 2 : 3 : 5 determine three angles.

Solution:

Let the common ratio be x The three angles are 2x, 3x and 5x 2x + 3x + 5x = 180° [Sum of the angles of triangle is 180°]

10x = 180°

x = 18010

2x = 2 ×18 = 36°

3x = 3 × 18 = 54°

5x = 5 × 18 = 90°

Question 6.

The angles of a triangle are arranged in ascending order of magnitude. If the difference between consecuttive angles is 15° find the three angles.

Solution:

Let the first angle be x then the second angle is x + 15 and third angle is x + 30.

x + x + 15 + x + 30 = 180° [Sum of the angles]

3x + 45 = 180

3x = 180 – 45

3x = 135

x = 1353 = 45°

First angle = x = 45°

Second angle = x + 15 = 45 + 15 = 60°

Third angle = x + 30 = 45 + 30 = 75°

Question 7.

The sum of two angles of a triangle is equal to its third angle. Determine the measure of third angle.

Solution:

Let the sum of two angles be x and the third angle be y

x + y = 180° [Sum of the angles of the triangle]

y + y = 180° [∴ sum of two angles = third angle]

2y = 180°

y = 1802

y = 90°

∴ The third angle is 90°

Question 9.

The angles of a triangle are x – 40°, x – 20 and 12 x + 15° find the value of x

Solution:

x – 40 + x – 20 + 12 x + 15 = 180° [Sum of the angles of a triangle]

x + x + 12 x – 60 + 15 = 1 80°

x + x + 12 x – 45 = 180°

2x + 12 x = 180 + 45

4x+x2 = 225

5x = 225 x 2

5x = 450

x = 4505

x = 90°

Question 10.

In ∆ABC ∠A – ∠B = 15° and ∠B – ∠C = 30° find ∆ABC ∠A , ∠B and ∠C

Solution:

∠A – ∠B = 15°

∴ ∠A = 15 + ∠B

∠B – ∠C = 30

∴ ∠B = 30 + ∠C

∠A + ∠B + ∠C = 180° [Sum of the angle of a triangle]

15 + ∠B + ∠B + ∠C = 180° (∴ ∠A = 15 + ∠B)

15 + 2∠B + ∠C = 180°

15 + 2[30 + ∠C + ∠C = 180° (∴ ∠B = 30 + ∠C)

15 + 60 + 2∠C] + ∠C = 180°

75 + 3∠C = 180°

3∠C = 180 – 75

3∠C = 105°

∠C = 105∘3

∠C = 35°

Now ∠B = 30 + ∠C

∠B = 30 + 35

∠B = 65°

∠A = 15 + ∠B

∠A = 15 + 65

∠A = 180°

Question 13.

In a triangle, each of the smaller angles is half of the largest angle. Find the angles.

Solution:

Let the smallest angle be x then the largest angle is 2x.

x + x + 2x = 180° [Sum of the angles of triangle]

4x = 180°

x = 1804

x = 45°

2x = 2 × 45° = 90°

∴ The angles are 45°, 45° and 90°

Question 14.

In a triangle each of the bigger angle is twice the third angle find the angles.

Solution:

Let the third angle be x. then the bigger angles is 2x 2x + 2x + x = 180° [Sum of the angles of triangle]

5x = 180°

x = 1805

x = 36°

2x = 2 × 36° = 72°

∴ The angles are 72°, 72° and 36°

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