Hard Level Worksheet Questions

Part A: Very Short Answer Questions (1 Mark Each)

This hard-level lines and angles worksheet challenges your understanding of complex angle relationships, algebraic angle problems, and advanced geometric reasoning.

First, let's solve advanced angle problems involving algebraic expressions and complex relationships.

1. Find the angle which is 4 times its complement.

Let the angle = x. Its complement = 90 - x90 + x.

Given: x = 4(90 - x)4+(90 - x)

So angle(x) = °

Awesome! The angle is 72°.

2. If two angles form a linear pair and one is 3x°, the other is (2x – 10)°, find x.

Simplifying x =

Great job! x = 38.

3. What is the angle which is equal to its supplement?

Supplementary angles are two angles whose measures add up to °.

So, angle = °

Perfect! Only 90° equals its supplement.

4. If ∠A = 2x – 15 and ∠B = x + 30 are vertically opposite angles, find x.

Simplifying x =

Excellent! x = 45.

5. Find an angle whose measure is one-fourth of its supplement.

Simplifying x = °

Super! The angle is 36°.

6. If two adjacent angles form a straight angle and one is 2/5 of the other, find both angles.

Let one angle = x, other = x

Simplifying x = °

The other angle = °

That's correct! The angles are 128.57° and 51.43°.

7. Two complementary angles are in the ratio 2:3. Find the angles.

Complementary angles are two angles whose measures add up to °.

Angles are ° and °

8. If the difference between two supplementary angles is 64°, find the angles.

Supplementary angles are two angles whose measures add up to °.

Let angles = x and .

x = °

Other angle = °

9. The supplement of an angle is 40° more than twice the angle. Find the angle.

Let angle = x.

By simplifying x = ° (upto 2 decimals)

You nailed it! The angle is 46.67°.

10. What type of angles are 90° and 90°, and what is their relationship?

They are Equal right anglesBoth are acute angles

Perfect! They are equal right angles.

Drag each angle relationship to its correct category:

Part B: Short Answer Questions (2 Marks Each)

1. Two adjacent supplementary angles are such that one is 5 times the other. Find both angles.

Let smaller angle = x, larger angle =

Since supplementary: x + 5x = °, so x = °

The angles are ° and °

Excellent! The angles are 30° and 150°.

2. ∠A and ∠B form a linear pair. ∠A is 20° less than twice ∠B. Find both angles.

Let ∠B = x, then ∠A =

Linear pair: x + (2x - 20) = °, so x = °

∠A = °

Great work! ∠B = 66.67° and ∠A = 113.33°.

3. If two vertically opposite angles are 6x – 5 and 4x + 15, find x and both angles.

Vertically opposite angles are : 6x - 5 = 4x + 15

Solving: x =

Both angles = °

Perfect! x = 10 and both angles are 55°.

4. A linear pair of angles are such that one angle is 15° more than its complement. Find both angles.

Let one angle = x. Its complement = .

Given: x=(90-x)+15x+15=(90-x)

Solving: x = °

Other angle = °

Outstanding! The angles are 52.5° and 127.5°.

5. ∠AOC and ∠BOC form a linear pair. If ∠AOC = 3x – 20 and ∠BOC = 2x + 10, find the angles.

Linear pair: (3x - 20) + (2x + 10) =

Simplifying: x =

∠AOC = °, ∠BOC = °

Fantastic! ∠AOC = 94° and ∠BOC = 86°.

Part C: Long Answer Questions (4 Marks Each)

1. Two lines intersect at point O forming four angles. One is (3x – 20)°, adjacent one is (2x + 40)°. Find all angles.

These angles form a linear pair because they are and sum to °

So, x =

First angle = 3(32) - 20 = °, second angle = 2(32) + 40 = °

The four angles are °, °, °, °

Excellent! The four angles are 76°, 104°, 76°, and 104°.

2. The supplement of an angle is 10° more than three times its complement. Find the angle.

Let angle = x. Complement = 90 - x180 - x, Supplement = 180 - x90 - x

Given: (180 - x) = 3(90 - x) + 10(180 - x) + 10 = 3(90 - x).

Solving: x = °

Verification: Complement = °, Supplement = °.

Check: 130 = 3() + 10 = °

Therefore, the angle is 50°.

3. Lines AB and CD intersect at O. If ∠AOC = (5x – 10)°, ∠BOD = (3x + 20)°, find all angles.

∠AOC and ∠BOD are vertically opposite: 5x - 10 =+- 3x + 20

Solving: x =

∠AOC = ∠BOD = °

Other pair: ∠AOD = ∠BOC = 180 - = ° each

Perfect! Two angles are 65° each and two are 115° each.

4. Two lines intersect forming eight angles. One angle is 2x°, adjacent is (x + 20)°.

Adjacent angles on straight line: 2x + (x + 20) = , so x =

The two angles are ° and °

All four angles: Two of ° and two of °

Fantastic! The angles are 106.67° and 73.33° (each pair appears twice).

Test your understanding with these challenging multiple choice questions:

For each question, click on the correct answer:

1. A straight angle is always equal to:

(a) 180° (b) 90° (c) 0° (d) 360°

Perfect! A straight angle always measures exactly 180°.

2. The supplement of an angle is 3 times the angle. What is the angle?

(a) 45° (b) 30° (c) 90° (d) 60°

Excellent! If 180-x = 3x, then 180 = 4x, so x = 45°.

3. Complementary angles add up to:

(a) 180° (b) 90° (c) 120° (d) 60°

Perfect! Complementary angles always sum to 90°.

4. If ∠x and ∠y are vertically opposite and ∠x = 2x – 20, ∠y = x + 10, then x is:

(a) 30 (b) 40 (c) 50 (d) 60

Great! 2x - 20 = x + 10, so x = 30.

5. What type of pair are two angles whose sum is 90° and are adjacent?

(a) Linear pair (b) Complementary and adjacent (c) Supplementary (d) None

Outstanding! They are complementary (sum = 90°) and adjacent (share a common side).

🎉 Exceptional Achievement! You've Mastered Hard-Level Lines and Angles!

Here's what you conquered:

• Complex algebraic angle relationships and equation solving
• Advanced problems involving ratios, complements, and supplements
• Multi-step angle calculations with vertically opposite angles
• Sophisticated linear pair and intersection problems
• Understanding impossible angle relationships
• Advanced geometric reasoning with multiple constraints
• Real-world applications of angle properties in complex figures

Your advanced understanding of lines and angles prepares you for coordinate geometry, trigonometry, and advanced geometric proofs!