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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