Solved Exercises on Types of Angles and Sides in Grade 4

Master types of angles and sides: acute, right, obtuse, straight, and reflex angles with 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: 30°, 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
30°
Acute
90°
Right
120°
Obtuse
180°
Straight
Angles
30°, 90°, 120°, 180°, 270°, 45°, 135°, 200°
Classify
A, R, O, S, Re, A, O, Re
Result
See below
Step 1: Analyze 30°

0° < 30° < 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

30°=A, 90°=R, 120°=O, 180°=S, 270°=Re, 45°=A, 135°=O, 200°=Re
Final answer:

30°=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 Triangle Side and Angle Classification
Exercise 2
A triangle has sides measuring 5 cm, 5 cm, and 8 cm. What type of triangle is this based on sides? What can you say about the angles in this triangle?
Definitions:

Equilateral triangle: All three sides equal

Isosceles triangle: Two sides equal

Scalene triangle: All sides different

5cm 5cm 8cm
Given
5cm, 5cm, 8cm
Compare
Two equal sides
Type
Isosceles
Step 1: Identify the sides

The triangle has sides measuring: 5 cm, 5 cm, and 8 cm

Step 2: Compare side lengths

Two sides are equal (5 cm = 5 cm), one is different (8 cm)

Step 3: Classify by sides

Since two sides are equal, this is an isosceles triangle

Step 4: Analyze the angles

In an isosceles triangle, the angles opposite equal sides are also equal

The angle opposite the 8 cm side will be different

Step 5: Estimate angle types

Since the base (8 cm) is longer than the equal sides (5 cm each), the triangle is acute

Isosceles triangle with two equal angles
Final answer:

This is an isosceles triangle with two equal sides (5 cm each). The triangle has two equal angles and one different angle.

Applied rules:

• Isosceles triangle: Two sides equal, two angles equal

• Side-angle relationship: Longer sides are opposite larger angles

• Triangle inequality: Sum of any two sides > third side

3 Quadrilateral Angle and Side Analysis
Exercise 3
A quadrilateral has all sides equal to 6 cm and all angles equal to 90°. What type of quadrilateral is this? What is its special name?
Definitions:

Square: A quadrilateral with all sides equal and all angles equal to 90°

Rhombus: A quadrilateral with all sides equal (but not necessarily 90° angles)

Rectangle: A quadrilateral with all angles equal to 90° (but not necessarily all sides equal)

6cm 90°
Sides
All 6cm
Angles
All 90°
Shape
Square
Step 1: Analyze the sides

All four sides measure 6 cm - they are all equal

Step 2: Analyze the angles

All four angles measure 90° - they are all right angles

Step 3: Apply quadrilateral definitions

A quadrilateral with all sides equal is a rhombus

A quadrilateral with all angles equal to 90° is a rectangle

A quadrilateral with both properties is a square

Step 4: Verify the classification

This shape meets both conditions: equal sides and right angles

Step 5: State the special name

The special name for this quadrilateral is a square

Square - special type of rectangle and rhombus
Final answer:

This is a square - a special quadrilateral with all sides equal and all angles equal to 90°

Applied rules:

• Square definition: Equal sides + right angles

• Hierarchy: Square is a special type of rectangle and rhombus

• Properties: All sides equal, all angles 90°, diagonals equal and perpendicular

Angle and Side Relationships
A + A = 180°, S₁ = S₂ → A₁ = A₂
Angle-Side Relationships
Triangle Sides
3 sides
Equilateral, Isosceles, Scalene
Quadrilateral Sides
4 sides
Square, Rectangle, Rhombus, Parallelogram
Angle-Side Rule
Longer side → Larger angle
Shorter side → Smaller angle
Triangle Angles
Sum = 180°
All angles add up to 180°
Triangle Types
By sides: Equilateral, Isosceles, Scalene
By angles: Acute, Right, Obtuse
Quadrilateral Types
Square, Rectangle, Rhombus, Trapezoid
Parallelogram, Kite
Side-Angle Relationship
Longest side ↔ Largest angle
Shortest side ↔ Smallest angle
Angle Sum Rules
Triangle: 180°
Quadrilateral: 360°
Solution: Exercises 4 to 5
4 Polygon Angle and Side Calculations
Exercise 4
A pentagon has five equal sides and five equal angles. What is the measure of each angle? What type of polygon is this?
Key information:

Regular polygon: A polygon with all sides and angles equal

Formula: Sum of interior angles = (n-2) × 180°

Regular Pentagon
Polygon
Pentagon (n=5)
Sum
(5-2)×180° = 540°
Each angle
540°÷5 = 108°
Step 1: Identify the polygon

A pentagon has 5 sides

Step 2: Apply the angle sum formula

Sum of interior angles = (n-2) × 180°

Sum = (5-2) × 180° = 3 × 180° = 540°

Step 3: Calculate each angle

Since all angles are equal: Each angle = 540° ÷ 5 = 108°

Step 4: Classify the polygon

Since all sides and angles are equal, this is a regular pentagon

Step 5: Verify the classification

Each angle (108°) is obtuse since 90° < 108° < 180°

Each angle = 108°, Regular pentagon
Final answer:

Each angle measures 108°, and this is a regular pentagon

Applied rules:

• Regular polygon: All sides and angles equal

• Interior angle sum: (n-2) × 180°

• Equal angles: Divide sum by number of sides

5 Complex Shape Analysis
Exercise 5
A hexagon has sides measuring 4 cm, 4 cm, 6 cm, 6 cm, 4 cm, and 6 cm. What can you say about this hexagon? How many different side lengths does it have?
Key information:

Hexagon: A polygon with 6 sides

Regular hexagon: All sides equal

Irregular hexagon: Sides of different lengths

4cm 4cm 6cm 6cm 4cm 6cm
Given
4, 4, 6, 6, 4, 6 cm
Unique
4 and 6 cm
Count
2 different lengths
Step 1: List all sides

Sides: 4 cm, 4 cm, 6 cm, 6 cm, 4 cm, 6 cm

Step 2: Identify unique lengths

Unique side lengths: 4 cm and 6 cm

Step 3: Count occurrences

4 cm occurs 3 times, 6 cm occurs 3 times

Step 4: Classify the hexagon

This is an irregular hexagon since not all sides are equal

Step 5: Describe the pattern

The hexagon has two different side lengths alternating or grouped

Irregular hexagon with 2 different side lengths
Final answer:

This is an irregular hexagon with 2 different side lengths (4 cm and 6 cm)

Applied rules:

• Regular polygon: All sides equal

• Irregular polygon: Sides of different lengths

• Classification: Based on side length patterns

Detailed Summary: Types of Angles and Sides
n = Number of sides, A + A = 180°
Polygon Properties
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°)

Equilateral triangle: A triangle with all three sides equal

Isosceles triangle: A triangle with two sides equal

Scalene triangle: A triangle with all sides different

Angle and side classification method:
  1. Count the number of sides: Determine the type of polygon
  2. Measure all sides: Compare side lengths to classify
  3. Measure all angles: Compare to 90° to classify
  4. Apply polygon properties: Use angle sum rules
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: Side-angle relationship: longer sides are opposite larger angles.
Common errors: Confusing acute with obtuse, forgetting that right angles are exactly 90°, misclassifying polygons based on sides vs angles.
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°

• Triangle angle sum: All three angles sum to 180°

• Quadrilateral angle sum: All four angles sum to 360°

• Polygon angle sum: (n-2) × 180°

Angle and Side Relationship Guide
Triangle
5cm 7cm 7cm
Longest side ↔ Largest angle
Square
All sides equal All angles 90°
Equal sides + angles
Pentagon
Regular 5 sides
Sum = 540°
Hexagon
Regular 6 sides
Sum = 720°

Shape Properties Summary: Remember the relationships between sides and angles:

  • Triangles: Longest side is opposite largest angle
  • Squares: All sides equal, all angles 90°
  • Regular polygons: All sides and angles equal
  • Angle sums: Triangle=180°, Quadrilateral=360°, Pentagon=540°, Hexagon=720°

Questions & Answers

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

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

  1. Acute = "A" for "All small": Think of "acute" as "a-cute" like something small and cute. Acute angles are small (less than 90°).
  2. Obtuse = "Obvious": Think of "obtuse" as "obvious" - these angles are obviously big and wide (greater than 90°).
  3. Size comparison: Acute angles are smaller than a right angle, obtuse angles are bigger than a right angle.

Another way to remember: "A" comes before "O" in the alphabet, just like acute angles (smaller) come before obtuse angles (larger) when comparing to 90°!

Practice by looking around for acute angles (like the tip of a slice of pizza) and obtuse angles (like the opening of a wide-open book).

Question: What happens if I measure an angle and get exactly 90°? Is it acute or right?

Answer: If an angle measures exactly 90°, it is definitely a right angle, not an acute angle!

Here's the important distinction:

  • Acute angles: Measure LESS than 90° (0° < acute angle < 90°)
  • Right angles: Measure EXACTLY 90° (right angle = 90°)

Think of it like this: acute angles are "less than" 90°, so they can't include 90° itself. Only right angles can equal exactly 90°.

This is why we have separate categories - each type has a specific range with exact boundaries. 90° is the boundary between acute and obtuse, but it belongs to the right angle category.

Remember: "Right on the dot!" - exactly 90° makes it a right angle.

Question: Can an angle be both acute and obtuse at the same time?

Answer: No, an angle cannot be both acute and obtuse at the same time! This would be impossible.

Think about the definitions:

  • Acute angle: Measures between 0° and 90° (less than 90°)
  • Obtuse angle: Measures between 90° and 180° (greater than 90°)

These ranges don't overlap at all! An angle can only have one measurement, so it can only fall into one category.

It's like asking if someone can be both shorter than 5 feet AND taller than 5 feet at the same time - it's impossible because these conditions contradict each other.

Every angle fits into exactly one category: acute, right, obtuse, straight, or reflex. This is called being "mutually exclusive" - the categories don't overlap.

Question: What's the difference between a straight angle and a reflex angle? They both seem pretty big.

Answer: Great observation! Both are large angles, but they differ in their exact measurements:

  • Straight angle: Measures exactly 180° - this is a perfectly flat line
  • Reflex angle: Measures greater than 180° but less than 360° - this is larger than a straight line but less than a full circle

Think of it this way:

A straight angle is like folding a piece of paper flat - you get a 180° angle.

A reflex angle is like continuing to fold past flat - now you have more than 180° but less than going all the way around (360°).

If you imagine a clock, a straight angle would be like the hands at 6 o'clock (180°), while a reflex angle would be like the hands at 8 o'clock if you measured the larger angle between them (240°).

The key is that reflex angles are "bigger than straight" but "smaller than full circle."

Question: How can I check if I've classified an angle correctly?

Answer: Here's a systematic way to verify your angle classification:

  1. Measure accurately: Use a protractor to get the precise angle measurement
  2. Compare to boundaries: Check where your measurement falls in relation to 90°, 180°, etc.
  3. Use the range definitions: Make sure your angle fits within the correct range for that type
  4. Visual check: Does your angle look like it should be acute (sharp), right (square corner), or obtuse (wide)?

Quick verification method:

  • Acute: Should look "pointy" and measure less than 90°
  • Right: Should look like a perfect corner and measure exactly 90°
  • Obtuse: Should look "wide" and measure more than 90° but less than 180°
  • Straight: Should look like a flat line and measure exactly 180°
  • Reflex: Should look like more than a flat line but less than a circle

Always double-check your protractor reading and make sure you're measuring from the correct starting point!