Introduction

In Key Stage 3, students learn to find missing angles in triangles, which is a fundamental skill in geometry. Triangles have specific properties that can be used to calculate unknown angles when given other angles. This skill helps students understand the relationship between angles and how to apply mathematical reasoning to solve problems involving geometric shapes.

Key Concepts

1. Sum of Angles in a Triangle: The three interior angles in any triangle always add up to $$ 180^\circ $$.
2. Types of Triangles:

• Equilateral Triangle: All three angles are equal, and each angle measures $$ 60^\circ $$.
• Isosceles Triangle: Two angles are equal, and the base angles are congruent.
• Scalene Triangle: All angles are different.

1. Exterior Angles: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
2. Supplementary Angles: When a straight line is created by two angles, those angles add up to $$ 180^\circ $$.

Using these properties, you can find unknown angles in various triangles.

Find Missing Angles in Triangles: Questions

Level 1: Easy

For each triangle, find the missing angle:

1. Triangle with angles $$ 70^\circ $$ and $$ 50^\circ $$.
2. Triangle with angles $$ 40^\circ $$ and $$ 80^\circ $$.
3. Triangle with angles $$ 30^\circ $$ and $$ 90^\circ $$.
4. Equilateral triangle (all angles are equal).
5. Triangle with angles $$ 45^\circ $$ and $$ 90^\circ $$.
6. Triangle with angles $$ 60^\circ $$ and $$ 60^\circ $$.
7. Triangle with angles $$ 50^\circ $$ and $$ 60^\circ $$.
8. Triangle with angles $$ 30^\circ $$ and $$ 70^\circ $$.
9. Triangle with angles $$ 80^\circ $$ and $$ 60^\circ $$.
10. Triangle with angles $$ 45^\circ $$ and $$ 45^\circ $$.

For each isosceles triangle, find the missing angle:

11. Isosceles triangle with one angle of $$ 70^\circ $$.
12. Isosceles triangle with one angle of $$ 50^\circ $$.
13. Isosceles triangle with one angle of $$ 40^\circ $$.
14. Isosceles triangle with one angle of $$ 60^\circ $$.
15. Isosceles triangle with one angle of $$ 80^\circ $$.
16. Isosceles triangle with one angle of $$ 55^\circ $$.
17. Isosceles triangle with one angle of $$ 65^\circ $$.
18. Isosceles triangle with one angle of $$ 45^\circ $$.
19. Isosceles triangle with one angle of $$ 30^\circ $$.
20. Isosceles triangle with one angle of $$ 85^\circ $$.

Level 2: Medium

For each triangle, find the missing angle:

1. Triangle with angles $$ 65^\circ $$ and $$ 85^\circ $$.
2. Triangle with angles $$ 120^\circ $$ and $$ 30^\circ $$.
3. Triangle with angles $$ 95^\circ $$ and $$ 40^\circ $$.
4. Triangle with angles $$ 55^\circ $$ and $$ 75^\circ $$.
5. Triangle with angles $$ 100^\circ $$ and $$ 45^\circ $$.
6. Triangle with angles $$ 110^\circ $$ and $$ 35^\circ $$.
7. Triangle with angles $$ 130^\circ $$ and $$ 20^\circ $$.
8. Triangle with angles $$ 125^\circ $$ and $$ 15^\circ $$.
9. Triangle with angles $$ 105^\circ $$ and $$ 40^\circ $$.
10. Triangle with angles $$ 135^\circ $$ and $$ 25^\circ $$.

For each isosceles triangle, find the missing angles:

11. Isosceles triangle with one angle of $$ 80^\circ $$.
12. Isosceles triangle with one angle of $$ 110^\circ $$.
13. Isosceles triangle with one angle of $$ 100^\circ $$.
14. Isosceles triangle with one angle of $$ 65^\circ $$.
15. Isosceles triangle with one angle of $$ 120^\circ $$.
16. Isosceles triangle with one angle of $$ 95^\circ $$.
17. Isosceles triangle with one angle of $$ 75^\circ $$.
18. Isosceles triangle with one angle of $$ 90^\circ $$.
19. Isosceles triangle with one angle of $$ 105^\circ $$.
20. Isosceles triangle with one angle of $$ 115^\circ $$.

Level 3: Hard

For each triangle, find the missing angle:

1. Triangle with angles $$ 85^\circ $$ and $$ 45^\circ $$.
2. Triangle with angles $$ 95^\circ $$ and $$ 50^\circ $$.
3. Triangle with angles $$ 105^\circ $$ and $$ 70^\circ $$.
4. Triangle with angles $$ 115^\circ $$ and $$ 40^\circ $$.
5. Triangle with angles $$ 125^\circ $$ and $$ 30^\circ $$.
6. Triangle with angles $$ 110^\circ $$ and $$ 20^\circ $$.
7. Triangle with angles $$ 130^\circ $$ and $$ 15^\circ $$.
8. Triangle with angles $$ 120^\circ $$ and $$ 55^\circ $$.
9. Triangle with angles $$ 135^\circ $$ and $$ 10^\circ $$.
10. Triangle with angles $$ 145^\circ $$ and $$ 25^\circ $$.

For each scalene triangle, find the missing angle:

11. Scalene triangle with angles $$ 115^\circ $$ and $$ 30^\circ $$.
12. Scalene triangle with angles $$ 140^\circ $$ and $$ 20^\circ $$.
13. Scalene triangle with angles $$ 130^\circ $$ and $$ 15^\circ $$.
14. Scalene triangle with angles $$ 100^\circ $$ and $$ 55^\circ $$.
15. Scalene triangle with angles $$ 135^\circ $$ and $$ 10^\circ $$.
16. Scalene triangle with angles $$ 120^\circ $$ and $$ 35^\circ $$.
17. Scalene triangle with angles $$ 110^\circ $$ and $$ 40^\circ $$.
18. Scalene triangle with angles $$ 105^\circ $$ and $$ 50^\circ $$.
19. Scalene triangle with angles $$ 125^\circ $$ and $$ 15^\circ $$.
20. Scalene triangle with angles $$ 145^\circ $$ and $$ 5^\circ $$.

Answers with Explanations

Level 1: Easy

For each triangle, find the missing angle:

1. Triangle with angles $$70^\circ$$ and $$50^\circ$$:
   The sum of angles in a triangle is $$180^\circ$$.
   $$180^\circ – (70^\circ + 50^\circ) = 60^\circ$$.
   Missing angle: $$60^\circ$$.
2. Triangle with angles $$40^\circ$$ and $$80^\circ$$:
   $$180^\circ – (40^\circ + 80^\circ) = 60^\circ$$.
   Missing angle: $$60^\circ$$.
3. Triangle with angles $$30^\circ$$ and $$90^\circ$$:
   $$180^\circ – (30^\circ + 90^\circ) = 60^\circ$$.
   Missing angle: $$60^\circ$$.
4. Equilateral triangle (all angles are equal):
   All angles in an equilateral triangle are $$60^\circ$$.
   Missing angle: $$60^\circ$$.
5. Triangle with angles $$45^\circ$$ and $$90^\circ$$:
   $$180^\circ – (45^\circ + 90^\circ) = 45^\circ$$.
   Missing angle: $$45^\circ$$.
6. Triangle with angles $$60^\circ$$ and $$60^\circ$$:
   $$180^\circ – (60^\circ + 60^\circ) = 60^\circ$$.
   Missing angle: $$60^\circ$$.
7. Triangle with angles $$50^\circ$$ and $$60^\circ$$:
   $$180^\circ – (50^\circ + 60^\circ) = 70^\circ$$.
   Missing angle: $$70^\circ$$.
8. Triangle with angles $$30^\circ$$ and $$70^\circ$$:
   $$180^\circ – (30^\circ + 70^\circ) = 80^\circ$$.
   Missing angle: $$80^\circ$$.
9. Triangle with angles $$80^\circ$$ and $$60^\circ$$:
   $$180^\circ – (80^\circ + 60^\circ) = 40^\circ$$.
   Missing angle: $$40^\circ$$.
10. Triangle with angles $$45^\circ$$ and $$45^\circ$$:
    $$180^\circ – (45^\circ + 45^\circ) = 90^\circ$$.
    Missing angle: $$90^\circ$$.

For each isosceles triangle, find the missing angle:

11. Isosceles triangle with one angle of $$70^\circ$$:
    In an isosceles triangle, the other two angles are equal.
    $$180^\circ – 70^\circ = 110^\circ$$.
    Each of the equal angles is $$\frac{110^\circ}{2} = 55^\circ$$.
    Each missing angle: $$55^\circ$$.
12. Isosceles triangle with one angle of $$50^\circ$$:
    $$180^\circ – 50^\circ = 130^\circ$$.
    Each of the equal angles is $$\frac{130^\circ}{2} = 65^\circ$$.
    Each missing angle: $$65^\circ$$.
13. Isosceles triangle with one angle of $$40^\circ$$:
    $$180^\circ – 40^\circ = 140^\circ$$.
    Each of the equal angles is $$\frac{140^\circ}{2} = 70^\circ$$.
    Each missing angle: $$70^\circ$$.
14. Isosceles triangle with one angle of $$60^\circ$$:
    $$180^\circ – 60^\circ = 120^\circ$$.
    Each of the equal angles is $$\frac{120^\circ}{2} = 60^\circ$$.
    Each missing angle: $$60^\circ$$.
15. Isosceles triangle with one angle of $$80^\circ$$:
    $$180^\circ – 80^\circ = 100^\circ$$.
    Each of the equal angles is $$\frac{100^\circ}{2} = 50^\circ$$.
    Each missing angle: $$50^\circ$$.
16. Isosceles triangle with one angle of $$55^\circ$$:
    $$180^\circ – 55^\circ = 125^\circ$$.
    Each of the equal angles is $$\frac{125^\circ}{2} = 62.5^\circ$$.
    Each missing angle: $$62.5^\circ$$.
17. Isosceles triangle with one angle of $$65^\circ$$:
    $$180^\circ – 65^\circ = 115^\circ$$.
    Each of the equal angles is $$\frac{115^\circ}{2} = 57.5^\circ$$.
    Each missing angle: $$57.5^\circ$$.
18. Isosceles triangle with one angle of $$45^\circ$$:
    $$180^\circ – 45^\circ = 135^\circ$$.
    Each of the equal angles is $$\frac{135^\circ}{2} = 67.5^\circ$$.
    Each missing angle: $$67.5^\circ$$.
19. Isosceles triangle with one angle of $$30^\circ$$:
    $$180^\circ – 30^\circ = 150^\circ$$.
    Each of the equal angles is $$\frac{150^\circ}{2} = 75^\circ$$.
    Each missing angle: $$75^\circ$$.
20. Isosceles triangle with one angle of $$85^\circ$$:
    $$180^\circ – 85^\circ = 95^\circ$$.
    Each of the equal angles is $$\frac{95^\circ}{2} = 47.5^\circ$$.
    Each missing angle: $$47.5^\circ$$.

Level 2: Medium

For each triangle, find the missing angle:

1. Triangle with angles $$65^\circ$$ and $$85^\circ$$:
   $$180^\circ – (65^\circ + 85^\circ) = 30^\circ$$.
   Missing angle: $$30^\circ$$.
2. Triangle with angles $$120^\circ$$ and $$30^\circ$$:
   $$180^\circ – (120^\circ + 30^\circ) = 30^\circ$$.
   Missing angle: $$30^\circ$$.
3. Triangle with angles $$95^\circ$$ and $$40^\circ$$:
   $$180^\circ – (95^\circ + 40^\circ) = 45^\circ$$.
   Missing angle: $$45^\circ$$.
4. Triangle with angles $$55^\circ$$ and $$75^\circ$$:
   $$180^\circ – (55^\circ + 75^\circ) = 50^\circ$$.
   Missing angle: $$50^\circ$$.
5. Triangle with angles $$100^\circ$$ and $$45^\circ$$:
   $$180^\circ – (100^\circ + 45^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
6. Triangle with angles $$110^\circ$$ and $$35^\circ$$:
   $$180^\circ – (110^\circ + 35^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
7. Triangle with angles $$130^\circ$$ and $$20^\circ$$:
   $$180^\circ – (130^\circ + 20^\circ) = 30^\circ$$.
   Missing angle: $$30^\circ$$.
8. Triangle with angles $$125^\circ$$ and $$15^\circ$$:
   $$180^\circ – (125^\circ + 15^\circ) = 40^\circ$$.
   Missing angle: $$40^\circ$$.
9. Triangle with angles $$105^\circ$$ and $$40^\circ$$:
   $$180^\circ – (105^\circ + 40^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
10. Triangle with angles $$135^\circ$$ and $$25^\circ$$:
    $$180^\circ – (135^\circ + 25^\circ) = 20^\circ$$.
    Missing angle: $$20^\circ$$.

For each isosceles triangle, find the missing angles:

11. Isosceles triangle with one angle of $$80^\circ$$:
    $$180^\circ – 80^\circ = 100^\circ$$.
    Each of the equal angles is $$\frac{100^\circ}{2} = 50^\circ$$. Each missing angle: $$50^\circ$$.
12. Isosceles triangle with one angle of $$110^\circ$$:
    $$180^\circ – 110^\circ = 70^\circ$$.
    Each of the equal angles is $$\frac{70^\circ}{2} = 35^\circ$$.
    Each missing angle: $$35^\circ$$.
13. Isosceles triangle with one angle of $$100^\circ$$:
    $$180^\circ – 100^\circ = 80^\circ$$.
    Each of the equal angles is $$\frac{80^\circ}{2} = 40^\circ$$.
    Each missing angle: $$40^\circ$$.
14. Isosceles triangle with one angle of $$65^\circ$$:
    $$180^\circ – 65^\circ = 115^\circ$$.
    Each of the equal angles is $$\frac{115^\circ}{2} = 57.5^\circ$$.
    Each missing angle: $$57.5^\circ$$.
15. Isosceles triangle with one angle of $$120^\circ$$:
    $$180^\circ – 120^\circ = 60^\circ$$.
    Each of the equal angles is $$\frac{60^\circ}{2} = 30^\circ$$.
    Each missing angle: $$30^\circ$$.
16. Isosceles triangle with one angle of $$95^\circ$$:
    $$180^\circ – 95^\circ = 85^\circ$$.
    Each of the equal angles is $$\frac{85^\circ}{2} = 42.5^\circ$$.
    Each missing angle: $$42.5^\circ$$.
17. Isosceles triangle with one angle of $$75^\circ$$:
    $$180^\circ – 75^\circ = 105^\circ$$.
    Each of the equal angles is $$\frac{105^\circ}{2} = 52.5^\circ$$.
    Each missing angle: $$52.5^\circ$$.
18. Isosceles triangle with one angle of $$90^\circ$$:
    $$180^\circ – 90^\circ = 90^\circ$$.
    Each of the equal angles is $$\frac{90^\circ}{2} = 45^\circ$$.
    Each missing angle: $$45^\circ$$.
19. Isosceles triangle with one angle of $$105^\circ$$:
    $$180^\circ – 105^\circ = 75^\circ$$.
    Each of the equal angles is $$\frac{75^\circ}{2} = 37.5^\circ$$.
    Each missing angle: $$37.5^\circ$$.
20. Isosceles triangle with one angle of $$115^\circ$$:
    $$180^\circ – 115^\circ = 65^\circ$$.
    Each of the equal angles is $$\frac{65^\circ}{2} = 32.5^\circ$$.
    Each missing angle: $$32.5^\circ$$.

Level 3: Hard

For each triangle, find the missing angle:

1. Triangle with angles $$85^\circ$$ and $$45^\circ$$:
   $$180^\circ – (85^\circ + 45^\circ) = 50^\circ$$.
   Missing angle: $$50^\circ$$.
2. Triangle with angles $$95^\circ$$ and $$50^\circ$$:
   $$180^\circ – (95^\circ + 50^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
3. Triangle with angles $$105^\circ$$ and $$70^\circ$$:
   $$180^\circ – (105^\circ + 70^\circ) = 5^\circ$$.
   Missing angle: $$5^\circ$$.
4. Triangle with angles $$115^\circ$$ and $$40^\circ$$:
   $$180^\circ – (115^\circ + 40^\circ) = 25^\circ$$.
   Missing angle: $$25^\circ$$.
5. Triangle with angles $$125^\circ$$ and $$30^\circ$$:
   $$180^\circ – (125^\circ + 30^\circ) = 25^\circ$$.
   Missing angle: $$25^\circ$$.
6. Triangle with angles $$110^\circ$$ and $$20^\circ$$:
   $$180^\circ – (110^\circ + 20^\circ) = 50^\circ$$.
   Missing angle: $$50^\circ$$.
7. Triangle with angles $$130^\circ$$ and $$15^\circ$$:
   $$180^\circ – (130^\circ + 15^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
8. Triangle with angles $$120^\circ$$ and $$55^\circ$$:
   $$180^\circ – (120^\circ + 55^\circ) = 5^\circ$$.
   Missing angle: $$5^\circ$$.
9. Triangle with angles $$135^\circ$$ and $$10^\circ$$:
   $$180^\circ – (135^\circ + 10^\circ) = 35^\circ$$.
   Missing angle: $$35^\circ$$.
10. Triangle with angles $$145^\circ$$ and $$25^\circ$$:
    $$180^\circ – (145^\circ + 25^\circ) = 10^\circ$$.
    Missing angle: $$10^\circ$$.

For each scalene triangle, find the missing angle:

11. Scalene triangle with angles $$115^\circ$$ and $$30^\circ$$:
    $$180^\circ – (115^\circ + 30^\circ) = 35^\circ$$.
    Missing angle: $$35^\circ$$.
12. Scalene triangle with angles $$140^\circ$$ and $$20^\circ$$:
    $$180^\circ – (140^\circ + 20^\circ) = 20^\circ$$.
    Missing angle: $$20^\circ$$.
13. Scalene triangle with angles $$130^\circ$$ and $$15^\circ$$:
    $$180^\circ – (130^\circ + 15^\circ) = 35^\circ$$.
    Missing angle: $$35^\circ$$.
14. Scalene triangle with angles $$100^\circ$$ and $$55^\circ$$:
    $$180^\circ – (100^\circ + 55^\circ) = 25^\circ$$.
    Missing angle: $$25^\circ$$.
15. Scalene triangle with angles $$135^\circ$$ and $$10^\circ$$:
    $$180^\circ – (135^\circ + 10^\circ) = 35^\circ$$.
    Missing angle: $$35^\circ$$.
16. Scalene triangle with angles $$120^\circ$$ and $$35^\circ$$:
    $$180^\circ – (120^\circ + 35^\circ) = 25^\circ$$.
    Missing angle: $$25^\circ$$.
17. Scalene triangle with angles $$110^\circ$$ and $$40^\circ$$:
    $$180^\circ – (110^\circ + 40^\circ) = 30^\circ$$.
    Missing angle: $$30^\circ$$.
18. Scalene triangle with angles $$105^\circ$$ and $$50^\circ$$:
    $$180^\circ – (105^\circ + 50^\circ) = 25^\circ$$.
    Missing angle: $$25^\circ$$.
19. Scalene triangle with angles $$125^\circ$$ and $$15^\circ$$:
    $$180^\circ – (125^\circ + 15^\circ) = 40^\circ$$.
    Missing angle: $$40^\circ$$.
20. Scalene triangle with angles $$145^\circ$$ and $$5^\circ$$:
    $$180^\circ – (145^\circ + 5^\circ) = 30^\circ$$.
    Missing angle: $$30^\circ$$.

This concludes the solutions and explanations for all the questions across the Easy, Medium, and Hard levels of finding missing angles in triangles. Each step follows the basic rule that the sum of interior angles in a triangle is always $$180^\circ$$, and the given known angles are subtracted from this total to find the missing value.