Solved Exercises on Types of Angles and Relationships in Grade 4

Master types of angles and their relationships: acute, right, obtuse, straight, and reflex angles through these 5 detailed exercises with visual diagrams.

Solution: Exercises 1 to 3
1 Angle Classification Practice
Exercise 1
Classify these angles as acute, right, obtuse, straight, or reflex: 25°, 90°, 120°, 180°, 270°, 45°, 135°, 200°
Definitions:

Acute: 0° < angle < 90°, Right: angle = 90°, Obtuse: 90° < angle < 180°

Straight: angle = 180°, Reflex: 180° < angle < 360°

Method:
  1. Compare each angle to the key values: 0°, 90°, 180°, 270°, 360°
  2. Determine which range the angle falls into
  3. Assign the appropriate classification
  4. Verify the classification is correct
25°
Acute
90°
Right
120°
Obtuse
180°
Straight
Angles
25°, 90°, 120°, 180°, 270°, 45°, 135°, 200°
Classify
A, R, O, S, Re, A, O, Re
Result
See below
Step 1: Analyze 25°

0° < 25° < 90° → Acute angle

Step 2: Analyze 90°

90° = 90° → Right angle

Step 3: Analyze 120°

90° < 120° < 180° → Obtuse angle

Step 4: Continue for all angles

180° = 180° → Straight angle

270°: 180° < 270° < 360° → Reflex angle

45°: 0° < 45° < 90° → Acute angle

135°: 90° < 135° < 180° → Obtuse angle

200°: 180° < 200° < 360° → Reflex angle

25°=A, 90°=R, 120°=O, 180°=S, 270°=Re, 45°=A, 135°=O, 200°=Re
Final answer:

25°=Acute, 90°=Right, 120°=Obtuse, 180°=Straight, 270°=Reflex, 45°=Acute, 135°=Obtuse, 200°=Reflex

Applied rules:

• Acute: 0° < angle < 90°

• Right: angle = 90°

• Obtuse: 90° < angle < 180°

• Straight: angle = 180°

• Reflex: 180° < angle < 360°

2 Adjacent Angles Relationship
Exercise 2
Two adjacent angles form a straight line. If one angle measures 75°, what is the measure of the other angle? What is the relationship between these angles?
Definitions:

Adjacent angles: Two angles that share a common vertex and side but no common interior points

Linear pair: Two adjacent angles that are supplementary (sum to 180°)

75° 105°
Given
One angle = 75°
Linear pair
Sum = 180°
Other angle
180° - 75° = 105°
Step 1: Understand the relationship

Two adjacent angles that form a straight line are called a linear pair

Step 2: Apply linear pair property

Linear pair angles are supplementary, meaning they sum to 180°

Step 3: Set up the equation

75° + other angle = 180°

Step 4: Solve for the unknown angle

Other angle = 180° - 75° = 105°

Step 5: Verify the relationship

75° + 105° = 180° ✓

Other angle = 105°, Linear pair (supplementary)
Final answer:

The other angle measures 105°, and the two angles form a linear pair (they are supplementary)

Applied rules:

• Linear pair: Adjacent angles that sum to 180°

• Supplementary angles: Two angles that sum to 180°

• Subtraction: To find one angle, subtract the known angle from 180°

3 Vertical Angles Relationship
Exercise 3
Two lines intersect forming four angles. If one angle measures 110°, what are the measures of the other three angles? Explain the relationships.
Definitions:

Vertical angles: Angles opposite each other when two lines intersect. They are equal.

Linear pair: Adjacent angles that sum to 180°

110° 70° 70° 110°
Given
One angle = 110°
Vertical
= 110°
Adjacent
= 70°
Step 1: Identify vertical angles

Vertical angles are equal, so the angle opposite to 110° is also 110°

Step 2: Find adjacent angles

Adjacent angles form a linear pair with the 110° angle

Adjacent angle = 180° - 110° = 70°

Step 3: Find the fourth angle

The fourth angle is vertical to the 70° angle, so it's also 70°

Step 4: Verify the sum

110° + 70° + 110° + 70° = 360° ✓

Step 5: Explain relationships

Vertical angles are equal, adjacent angles are supplementary

Angles: 110°, 70°, 110°, 70°
Final answer:

The four angles are 110°, 70°, 110°, 70°. The relationships are: vertical angles are equal and adjacent angles are supplementary.

Applied rules:

• Vertical angles: Opposite angles are equal

• Linear pairs: Adjacent angles sum to 180°

• Circle sum: Four angles around a point sum to 360°

Angle Relationships Guide
A + A = 180°, V = V, C + C = 90°
Angle Relationships
Complementary Angles
∠A + ∠B = 90°
Sum to 90°
Supplementary Angles
∠A + ∠B = 180°
Sum to 180°
Vertical Angles
∠A = ∠B
Opposite angles are equal
Linear Pair
∠A + ∠B = 180°
Adjacent supplementary angles
Complementary
Two angles that sum to 90°
Example: 30° + 60° = 90°
Supplementary
Two angles that sum to 180°
Example: 70° + 110° = 180°
Vertical
Opposite angles are equal
Example: 120° = 120°
Linear Pair
Adjacent angles that sum to 180°
Example: 130° + 50° = 180°
Solution: Exercises 4 to 5
4 Complementary and Supplementary Angles
Exercise 4
Find the complement and supplement of an angle that measures 35°. Explain the difference between complementary and supplementary angles.
Definitions:

Complement: An angle that adds to the original angle to make 90°

Supplement: An angle that adds to the original angle to make 180°

35°
Angle A (35°)
55°
Complement (55°)
Given
35°
Complement
90° - 35° = 55°
Supplement
180° - 35° = 145°
Step 1: Find the complement

Complement = 90° - 35° = 55°

Check: 35° + 55° = 90° ✓

Step 2: Find the supplement

Supplement = 180° - 35° = 145°

Check: 35° + 145° = 180° ✓

Step 3: Explain the difference

Complementary angles sum to 90° (like corners), supplementary angles sum to 180° (like straight lines)

Step 4: Verify both relationships

Both 35° + 55° = 90° and 35° + 145° = 180° are correct

Complement = 55°, Supplement = 145°
Final answer:

The complement of 35° is 55°, and the supplement of 35° is 145°

Applied rules:

• Complementary: Two angles that sum to 90°

• Supplementary: Two angles that sum to 180°

• Arithmetic: Subtraction to find missing angle

5 Angle Relationships in Geometry
Exercise 5
Three angles form a straight line. The first angle is twice the size of the second angle, and the third angle is 30° more than the second angle. Find the measure of each angle.
Key information:

Three angles on a straight line sum to 180°

2x° x° (x+30)°
Define
Second = x, First = 2x, Third = x+30
Equation
2x + x + (x+30) = 180°
Solve
4x + 30 = 180°, x = 37.5°
Step 1: Define variables

Let the second angle = x°

First angle = 2x° (twice the second)

Third angle = (x + 30)° (30° more than second)

Step 2: Set up the equation

Angles on a straight line sum to 180°:

2x + x + (x + 30) = 180°

Step 3: Simplify and solve

2x + x + x + 30 = 180°

4x + 30 = 180°

4x = 150°

x = 37.5°

Step 4: Find each angle

First angle: 2x = 2 × 37.5° = 75°

Second angle: x = 37.5°

Third angle: x + 30 = 37.5° + 30° = 67.5°

Step 5: Verify the solution

75° + 37.5° + 67.5° = 180° ✓

Angles: 75°, 37.5°, 67.5°
Final answer:

The three angles measure 75°, 37.5°, and 67.5°

Applied rules:

• Straight line: Angles sum to 180°

• Algebra: Use variables to represent unknowns

• Relationships: Translate words into mathematical expressions

Detailed Summary: Types of Angles and Relationships
A < R < O < S < Re
Angle Size Order
Key definitions:

Acute angle: An angle that measures greater than 0° but less than 90° (0° < angle < 90°)

Right angle: An angle that measures exactly 90° (angle = 90°)

Obtuse angle: An angle that measures greater than 90° but less than 180° (90° < angle < 180°)

Straight angle: An angle that measures exactly 180° (angle = 180°)

Reflex angle: An angle that measures greater than 180° but less than 360° (180° < angle < 360°)

Angle classification method:
  1. Measure the angle: Use a protractor to find the exact degree measure
  2. Compare to reference angles: Compare to 0°, 90°, 180°, 270°, 360°
  3. Identify the range: Determine which category the angle falls into
  4. Classify: Assign the appropriate angle type
Tip 1: Remember "A Cute" for acute angles - they're small and cute, less than 90°.
Tip 2: "Right on!" - right angles are exactly 90°, just right!
Tip 3: "Obvious" - obtuse angles are obvious because they're larger than 90°.
Tip 4: "Straight line" - straight angles form a perfectly flat line at 180°.
Common errors: Confusing acute with obtuse, forgetting that right angles are exactly 90°, misclassifying angles near boundaries.
Memory aid: "A ROSR" - Acute (0-90), Right (90), Obtuse (90-180), Straight (180), Reflex (180-360).
Essential facts to remember:

• Acute: 0° < angle < 90°

• Right: angle = 90°

• Obtuse: 90° < angle < 180°

• Straight: angle = 180°

• Reflex: 180° < angle < 360°

• Classification: Each angle fits into exactly one category

Angle Relationship Visual Guide
Complementary
30° 60°
30° + 60° = 90°
Supplementary
70° 110°
70° + 110° = 180°
Vertical
120° 120° 60° 60°
Opposite = Equal
Linear Pair
130° 50°
Adjacent = 180°

Angle Relationships Summary: Remember these important angle relationships:

  • Complementary angles form a corner-like shape and sum to 90°
  • Supplementary angles form a straight line and sum to 180°
  • Vertical angles are opposite each other when lines cross and are equal
  • Linear pairs are adjacent angles that form a straight line and sum to 180°

Questions & Answers

Question: How do I remember the difference between complementary and supplementary angles? They both sound similar.

Answer: Great question! Here are some helpful memory tricks:

  1. Complementary starts with C - like Corner. Think of a corner of a square, which is 90°.
  2. Supplementary starts with S - like Straight. Think of a straight line, which is 180°.
  3. Also, C comes before S in the alphabet, just like 90° comes before 180° in number order!

So remember: Complementary = Corner (90°) and Supplementary = Straight (180°).

Practice by looking around for 90° corners (complementary) and 180° straight lines (supplementary) to reinforce the concepts!

Question: When two lines cross, why are the opposite angles always equal?

Answer: These opposite angles are called vertical angles, and they're always equal due to a mathematical property!

Think about it this way: when two lines intersect, they form four angles around the point where they meet. The angles next to each other (adjacent) are supplementary (they add up to 180°).

If angle A and angle B are adjacent and add to 180°, and angle A and angle C (across from angle B) are also adjacent and add to 180°, then angle B must equal angle C!

This is because both angle B and angle C are what's needed to make angle A equal 180°. Since they're both "what's needed," they must be equal!

It's like if you and your friend both need the same amount of money to buy the same toy, you both have the same amount!

Question: How do I know if angles are adjacent? Sometimes I think they are, but I'm wrong.

Answer: Adjacent angles have to meet three specific requirements:

  1. Share the same vertex: They have the same corner point
  2. Share a common side: They share one ray (the line that goes out from the corner)
  3. No overlapping interiors: They don't overlap inside - there's no shared area between them

Think of it like two slices of pizza that share the same center point and one edge of crust, but don't overlap on the pizza surface.

A good way to check: Put your finger on the vertex (corner), trace along the shared side, and see if you can go to both angles without lifting your finger from the shared part.

If you can, they might be adjacent. If there's any overlap in the "inside" parts of the angles, they're not adjacent.

Question: What's the difference between a linear pair and just supplementary angles?

Answer: Great observation! There is a key difference:

  • Supplementary angles: Any two angles that add up to 180°, regardless of their position
  • Linear pair: Two angles that are BOTH adjacent AND supplementary (they form a straight line)

So all linear pairs are supplementary angles, but not all supplementary angles are linear pairs!

Think of it this way: A linear pair is a special type of supplementary angle pair that has the extra requirement of being adjacent (sharing a vertex and side).

Example: Two separate angles that each measure 90° are supplementary (90° + 90° = 180°), but they're not a linear pair because they're not adjacent. However, two angles that form a straight line are both supplementary AND a linear pair.

Question: How can I check if my answers are correct when solving angle relationship problems?

Answer: Here are several ways to verify your angle relationship solutions:

  1. Check the defining relationship:
    • For complementary: Do the angles add up to 90°?
    • For supplementary: Do the angles add up to 180°?
    • For vertical: Are opposite angles equal?
  2. Draw a diagram: Sketch the situation to visually confirm your answer makes sense
  3. Use logical reasoning: Does your answer fit the context of the problem?
  4. Plug back in: Substitute your answer back into the original conditions

For example, if you found that two complementary angles are 35° and 55°, check: 35° + 55° = 90° ✓

Always take a moment to verify your work - it's much easier to catch small mistakes early than to fix them later!