Solved Exercises on Angles in Polygons in Grade 3

Master angle properties: triangles, quadrilaterals, angle sums, and geometric relationships through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Triangle Angle Sum
Exercise 1
In a triangle, two angles measure 50° and 60°. What is the measure of the third angle?
Definition:

Triangle angle sum: The sum of all three interior angles in any triangle is always 180°.

Solution method:
  1. Add the known angles
  2. Subtract from the total (180°)
  3. Find the missing angle
Known angles
50° + 60°
Sum
110°
Missing angle
180° - 110° = 70°
Step 1: Identify the rule

Triangle angle sum = 180°

Step 2: Add known angles

50° + 60° = 110°

Step 3: Find the missing angle

Third angle = 180° - 110° = 70°

Third angle = 70°
Final answer:

The third angle measures 70° because 50° + 60° + 70° = 180°.

Applied rules:

• Triangle angle sum: Always equals 180°

• Subtraction principle: Missing value = Total - Known values

• Verification: Check that all angles add up to 180°

2 Rectangle Angles
Exercise 2
A rectangle has four angles. What is the measure of each angle? What is their total sum?
Definition:

Rectangle: A quadrilateral with 4 right angles (each 90°). Quadrilateral angle sum: Always 360°.

Rectangle angles
4 × 90°
Total sum
360°
Step 1: Identify rectangle properties

Every rectangle has 4 right angles, each measuring 90°

Step 2: Calculate total

4 angles × 90° each = 360°

Step 3: Verify with quadrilateral rule

All quadrilaterals have angle sum of 360° ✓

Each angle = 90°, Total = 360°
Final answer:

Each angle measures 90°. The total sum of all four angles is 360°.

Applied rules:

• Rectangle definition: 4 right angles (90° each)

• Quadrilateral angle sum: Always equals 360°

• Right angle measure: Exactly 90°

3 Pentagon Angle Calculation
Exercise 3
A pentagon has 5 equal angles. What is the measure of each angle?
Definition:

Pentagon: A polygon with 5 sides. Pentagon angle sum: (n-2) × 180° = (5-2) × 180° = 540°.

Pentagon angle sum
(5-2) × 180° = 540°
Each angle
540° ÷ 5 = 108°
Step 1: Calculate total angle sum

For any polygon: (n-2) × 180° where n = number of sides

For pentagon: (5-2) × 180° = 3 × 180° = 540°

Step 2: Divide equally among angles

If all 5 angles are equal: 540° ÷ 5 = 108° per angle

Step 3: Verify calculation

5 angles × 108° each = 540° ✓

Each angle = 108°
Final answer:

Each angle measures 108° because the total sum of 540° is divided equally among 5 angles.

Applied rules:

• Polygon angle sum formula: (n-2) × 180°

• Division principle: Equal parts = Total ÷ Number of parts

• Regular polygon: All angles are equal

Rules and methods, laws,...
(n-2) × 180°
Polygon Angle Sum Formula
Triangle
Sum = 180°
3 sides
Quadrilateral
Sum = 360°
4 sides
Pentagon
Sum = 540°
5 sides
Key definitions:

Angle: The space between two intersecting lines measured in degrees (°)

Right angle: An angle measuring exactly 90°

Acute angle: An angle measuring less than 90°

Obtuse angle: An angle measuring more than 90° but less than 180°

Complete methodology:
  1. Identify polygon type: Count the number of sides
  2. Apply angle sum formula: (n-2) × 180°
  3. Calculate individual angles: If equal, divide total by number of angles
  4. Verify: Check that all angles add up to the expected sum
Tip 1: Remember: Triangle = 180°, Quadrilateral = 360°, Pentagon = 540°.
Tip 2: Use a protractor to measure angles in physical shapes.
Tip 3: For regular polygons: each angle = total sum ÷ number of angles.
Tip 4: Always verify: add all angles to check they equal the expected sum.
Common errors: Forgetting angle sum formulas, confusing acute/obtuse/right angles, miscalculating missing angles.
Exam preparation: Memorize angle sums for triangles and quadrilaterals, practice the formula, know angle types.
Properties to know by heart:

• Triangle: 3 angles, sum = 180°

• Quadrilateral: 4 angles, sum = 360°

• Pentagon: 5 angles, sum = 540°

• Hexagon: 6 angles, sum = 720°

• General formula: (n-2) × 180° for n-sided polygon

Detailed Summary: Angles in Polygons

Definitions

Angle: The amount of turn between two lines that meet at a point, measured in degrees (°).

Right angle: An angle of exactly 90° (like the corner of a square).

Acute angle: An angle less than 90° (smaller than a right angle).

Obtuse angle: An angle greater than 90° but less than 180° (larger than a right angle).

Polygon: A closed shape made of straight lines with interior angles.

Core Rules

• Triangle angle sum: 180°

• Quadrilateral angle sum: 360°

• Polygon angle sum: (n-2) × 180° where n = number of sides

• All angles in a polygon must add up to the total sum

• In regular polygons, all angles are equal

Calculation Methods

1. Count the number of sides (n)

2. Apply formula: (n-2) × 180°

3. For equal angles: divide total by number of angles

4. For missing angles: subtract known angles from total

5. Verify by adding all angles

Examples

Triangle: 3 sides → (3-2) × 180° = 180°

Square: 4 sides → (4-2) × 180° = 360° (each 90°)

Regular Pentagon: 5 sides → (5-2) × 180° = 540° (each 108°)

Regular Hexagon: 6 sides → (6-2) × 180° = 720° (each 120°)

Tips & Tricks

• Remember: Triangle = 180°, Quadrilateral = 360°

• Each additional side adds 180° to the total

• Use the "n-2" pattern: 3→1, 4→2, 5→3, etc.

• Draw lines from one vertex to create triangles

• Always check: do angles add up correctly?

Key Takeaways

• The angle sum depends only on the number of sides

• The formula (n-2) × 180° works for any polygon

• Regular polygons have equal angles

• Always verify your calculations

• Practice with different polygon types

Additional Learning Materials
(n-2) × 180°
General Angle Sum Formula
Advanced definitions:

Exterior angle: The angle formed outside a polygon when extending one side

Central angle: An angle whose vertex is at the center of a regular polygon

Supplementary angles: Two angles that add up to 180°

Advanced calculations:
  1. Individual angle in regular polygon: (n-2) × 180° ÷ n
  2. Exterior angle sum: Always 360° for any polygon
  3. Individual exterior angle: 360° ÷ n
  4. Interior + exterior: Always supplementary (add to 180°)
Advanced Tip 1: Interior and exterior angles are supplementary (add to 180°).
Advanced Tip 2: The exterior angle sum is always 360°, regardless of the number of sides.
Advanced Tip 3: As the number of sides increases, each interior angle gets closer to 180°.
Advanced Tip 4: The central angle of a regular polygon is 360° ÷ n.
Real-world connections: Architecture uses angle properties for stability, art uses polygons for design.
Advanced applications: Engineering, construction, and computer graphics rely on polygon angle properties.
Additional formulas:

• Individual interior angle: (n-2) × 180° ÷ n

• Individual exterior angle: 360° ÷ n

• Number of triangles formed: n-2

• Central angle: 360° ÷ n

• Sum of exterior angles: Always 360°

Questions & Answers

Question: Why does the angle sum formula (n-2) × 180° work? Can you explain it simply?

Answer: This formula works because we can break any polygon into triangles! Here's the simple explanation:

  • Take any polygon and pick one corner (vertex)
  • Draw lines from that corner to all other corners (except the two adjacent ones)
  • You'll create triangles inside the polygon
  • The number of triangles = n-2 (where n is the number of sides)
  • Each triangle contributes 180° to the total
  • So total = (n-2) × 180°

For example: A quadrilateral (4 sides) can be split into 2 triangles, so total = 2 × 180° = 360°

This visual approach helps you understand why the formula works!

Question: How do I know if an angle is acute, right, or obtuse? It's hard to tell sometimes.

Answer: Here's a simple way to remember and identify different angle types:

  • Acute angle: Smaller than a right angle (less than 90°) - "sharp" and narrow
  • Right angle: Exactly 90° - looks like the corner of a square
  • Obtuse angle: Larger than a right angle but smaller than a straight line (between 90° and 180°)

Memory tricks:

  • "Acute" sounds like "cute" - small and cute angles
  • "Right" means correct - 90° is the "right" angle
  • "Obtuse" sounds like "obvious" - the angle is obviously big

You can also compare to the corner of a piece of paper - if it fits perfectly, it's 90°. If it's smaller, it's acute. If it's bigger, it's obtuse.

Question: What's the difference between interior and exterior angles? I keep getting confused.

Answer: Here's the key difference:

  • Interior angle: The angle inside the polygon, between two adjacent sides
  • Exterior angle: The angle outside the polygon, formed by extending one side

Think of walking around a polygon:

  • At each corner, you turn through the exterior angle
  • After going around completely, you've turned a full circle = 360°
  • That's why the sum of exterior angles is always 360°

Important relationship: Interior + Exterior = 180° (they are supplementary angles)

For example, in a regular hexagon: Interior = 120°, Exterior = 60°, and 120° + 60° = 180°

Question: How can I measure angles in real life? Do I need special tools?

Answer: Yes, you can measure angles with tools, but here are different approaches:

  • Protractor: The standard tool for measuring angles in degrees
  • Angle estimation: Compare to known angles (right angle = 90°)
  • Pattern recognition: Identify common angle measures

How to use a protractor:

  • Place the center of the protractor on the angle's vertex
  • Line up one side of the angle with the baseline of the protractor
  • Read the degree measure where the other side crosses the scale

For estimation: Use the corner of a piece of paper as a 90° reference, then estimate if the angle is smaller (acute) or larger (obtuse).

Practice with everyday objects: corners of books, walls, doors, and windows!

Question: Where do I see angles in polygons in real life? This helps me understand better!

Answer: Great question! Here are real-life examples of polygon angles:

  • Triangles: Roof angles, pyramid faces, traffic signs - all have 180° total
  • Squares/Rectangles: Window corners, book corners, room corners - all have 90° angles
  • Stop signs (octagons): Each angle is about 135°
  • Honeycomb (hexagons): Each angle is 120°
  • Floor tiles: Different polygons with various angle combinations

Try this activity: Measure the angles in your bedroom! Check door frames, window corners, and any rectangular objects. You'll see that rectangles always have four 90° angles.

Architecture and construction heavily rely on angle measurements for stability and design!