Solved Exercises on Angles Measured in Degrees in Grade 4

Master angle measurement in degrees: comparing angle sizes, converting measurements, and practical applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing angle sizes
Exercise 1
Which angle is larger: ∠ABC measuring 65° or ∠DEF measuring 82°? How much larger is it?
Definition:

Degree measure: The unit used to measure angles, where larger numbers represent wider angles

Comparison method:
  1. Compare the degree measures directly
  2. The angle with the larger number of degrees is wider
  3. To find the difference, subtract the smaller from the larger
∠ABC
65°
∠DEF
82°
Difference
82° - 65° = 17°
Step 1: Compare the degree measures

65° vs 82°

Step 2: Determine which is larger

82° > 65°, so ∠DEF is larger

Step 3: Calculate the difference

82° - 65° = 17°

Step 4: State the answer

∠DEF is larger by 17°

∠DEF (82°) is larger by 17°
Final answer:

∠DEF measuring 82° is larger than ∠ABC measuring 65° by 17°.

Applied rules:

• Comparison: Higher degree measure means wider angle

• Difference: Subtract smaller from larger to find difference

• Units: Always include degree symbol (°) in final answer

2 Converting angle measurements
Exercise 2
A right angle measures 90°. Express this as a fraction of a full circle (360°). Then convert this to a percentage.
Definition:

Full circle: A complete rotation around a point measures 360°

Right angle
90°
Full circle
360°
Fraction
1/4
Step 1: Write as a fraction

Right angle / Full circle = 90° / 360°

Step 2: Simplify the fraction

90° / 360° = 1/4 (divide both by 90)

Step 3: Convert to percentage

1/4 × 100% = 25%

Step 4: Verify the calculation

25% of 360° = 0.25 × 360° = 90° ✓

Right angle = 1/4 of a circle = 25%
Final answer:

A right angle (90°) is 1/4 of a full circle (360°), which equals 25%.

Applied rules:

• Fraction: Part / Whole = Angle measure / 360°

• Simplification: Divide numerator and denominator by GCD

• Percentage: Fraction × 100%

3 Real-world angle application
Exercise 3
A pizza is cut into 8 equal slices. What is the angle measure of each slice? If you eat 3 slices, what angle of the pizza have you consumed?
Definition:

Equal parts: When a circle is divided equally, each part has the same angle measure

Total angle
360°
Number of slices
8
Per slice
45°
Step 1: Calculate angle per slice

Total angle ÷ Number of slices = 360° ÷ 8 = 45° per slice

Step 2: Calculate angle for 3 slices

Angle per slice × Number of slices eaten = 45° × 3 = 135°

Step 3: Verify the calculation

45° × 8 = 360° ✓ (for all slices)

135° + remaining angle = 135° + (360° - 135°) = 360° ✓

Step 4: State both answers

Each slice = 45°; 3 slices = 135°

Each slice: 45°, 3 slices: 135°
Final answer:

Each slice measures 45°. If you eat 3 slices, you consume 135° of the pizza.

Applied rules:

• Equal division: Total angle ÷ Number of parts

• Multiplication: Angle per part × Number of parts

• Verification: Check that parts sum to total

Rules and methods, laws,...
1° = 1/360 of a circle
Degree Definition
Right Angle
90°
Quarter of a circle
Straight Angle
180°
Half of a circle
Full Circle
360°
Complete rotation

Degree Measurements

📐
Full Circle
A complete rotation around a point measures 360°. This is the basis for all angle measurements.
Example: Earth's rotation = 360°
Right Angle
A quarter of a circle measures 90°. This is a common reference angle found in squares and rectangles.
Example: Corner of a book = 90°
Straight Angle
Half of a circle measures 180°. This forms a straight line.
Example: Flat surface = 180°
Measuring Angles
1
Identify Vertex
2
Position Protractor
3
Read Measurement
4
Record Degrees
Key Concepts & Practice Tips
Fraction = Part/Whole
Angle Calculations
Key definitions:

Degree (°): The unit used to measure angles

Vertex: The point where two rays meet to form an angle

Ray: A line that starts at a point and extends infinitely in one direction

Protractor: A tool used to measure angles in degrees

Acute angle: Less than 90°

Obtuse angle: More than 90° but less than 180°

Complete methodology:
  1. Identify the problem: Determine what is being asked (compare, calculate, measure)
  2. Select the approach: Choose the appropriate method (comparison, division, etc.)
  3. Perform calculations: Apply mathematical operations accurately
  4. Verify results: Check that your answer makes sense
  5. Express clearly: Include units and provide a complete answer
Tip 1: Remember: the bigger the degree number, the wider the angle.
Tip 2: A right angle (90°) is like the corner of a square.
Tip 3: Always include the degree symbol (°) when writing angle measurements.
Tip 4: When dividing a circle into equal parts, divide 360° by the number of parts.
Common errors: Confusing angle sizes, forgetting degree symbol, miscalculating fractions.
Exam preparation: Practice comparisons, memorize key angle measures, work on fraction calculations.
Facts to remember:

• A full circle measures 360°

• A right angle measures 90°

• A straight angle measures 180°

• An acute angle is less than 90°

• An obtuse angle is greater than 90° but less than 180°

Comprehensive Summary: Angles Measured in Degrees

This summary covers all essential concepts, rules, and methods for understanding angles measured in degrees.

Definitions

Degree (°): The unit used to measure angles, with 360° forming a complete circle.

Angle: The amount of turn between two rays that share a common endpoint (vertex).

Vertex: The point where two rays meet to form an angle.

Ray: A line that starts at a point and extends infinitely in one direction.

Protractor: A semicircular tool used to measure angles in degrees.

Core Rules

Circle measurement: A complete rotation is 360°.

Right angle: Measures exactly 90°.

Straight angle: Measures exactly 180°.

Acute angle: Measures less than 90°.

Obtuse angle: Measures greater than 90° but less than 180°.

Methods & Procedures

Comparing angles: Higher degree measure indicates a wider angle.

Converting to fractions: Part ÷ Whole = Fraction of circle.

Calculating equal parts: Total degrees ÷ Number of parts.

Percentage calculation: Fraction × 100%.

Examples

Simple: 45° < 90°, so 45° is an acute angle.

Intermediate: 120° is obtuse since it's between 90° and 180°.

Advanced: Pizza cut into 6 pieces = 360° ÷ 6 = 60° per slice.

Tips & Tricks

Memory aid: "Right" angle = "Right" corner of a square (90°).

Visualization: Think of a clock face for angle estimation.

Verification: Check that equal parts multiply to total degrees.

Common Mistakes

Unit confusion: Forgetting to write the degree symbol (°).

Size confusion: Thinking smaller numbers mean smaller angles (they do!)

Calculation errors: Arithmetic mistakes in division or multiplication.

Questions & Answers

Question: Why did people decide to make a full circle 360 degrees? Why not 100 or 1000 degrees?

Answer: The ancient Babylonians chose 360° for a full circle around 3000 BC. They had good reasons:

  • They used a base-60 counting system (sexagesimal), and 360 is easily divisible by many numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180.
  • It was close to the number of days in a year (365), which helped with astronomical calculations.
  • It divides evenly into 6 equilateral triangles, which was important in early geometry.

While 100 or 1000 might seem more familiar today, 360° has been used consistently for thousands of years and is now the standard worldwide.

Interestingly, some mathematicians use radians instead of degrees for advanced calculations!

Question: How can I estimate angle measures without a protractor? Sometimes I need to make a quick guess.

Answer: Here are some useful estimation techniques:

  • Reference angles: Memorize key angles - 90° (corner of a square), 45° (half of a right angle), 180° (straight line).
  • Hand method: Spread your fingers wide - the angle between thumb and pinky is roughly 90°.
  • Comparison: Compare to known angles - is it wider or narrower than a right angle?
  • Division: If an angle looks like it's about 1/3 of a right angle, that's about 30°.

Practice with common objects: the angle of a door partially open, the hands of a clock at different times, or the corner of a book.

With practice, your estimates will become more accurate!

Question: What's the difference between an acute angle and an obtuse angle? They both seem pretty wide to me.

Answer: The key is to compare both to a right angle (90°):

  • Acute angle: Less than 90° (smaller than a right angle) - think of the sharp point of a slice of pizza.
  • Obtuse angle: Greater than 90° but less than 180° (wider than a right angle but not quite flat) - think of an open book that's more than halfway open.

Here are some examples to help you visualize:

  • Acute: 30° (the angle of a slice of pie), 60° (the angle of a triangular ruler)
  • Obtuse: 120° (the angle between two hands of a clock at 4 o'clock), 150° (wide open scissors)

Remember: Acute angles are "cute" because they're small, while obtuse angles are "obvious" because they're large!