Solved Exercises on Angle Word Problems in Grade 4

Master angle word problems: identifying angles, solving angle relationships, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding missing angles in a triangle
Exercise 1
In a triangle, one angle measures 50° and another measures 60°. What is the measure of the third angle?
Definition:

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

Solution method:
  1. Add the known angles together
  2. Subtract the sum from 180°
  3. The result is the missing angle
Known angles
50° + 60°
Sum
110°
Missing angle
70°
Step 1: Add the known angles

50° + 60° = 110°

Step 2: Subtract from 180°

180° - 110° = 70°

Step 3: Verify the answer

50° + 60° + 70° = 180° ✓

The third angle measures 70°
Final answer:

The third angle measures 70°

Applied rules:

• Triangle Sum Theorem: All triangles have angles that sum to 180°

• Subtraction: Missing angle = 180° - (sum of known angles)

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

2 Complementary angles
Exercise 2
Two angles are complementary. One angle measures 35°. What is the measure of the other angle?
Definition:

Complementary Angles: Two angles whose sum equals 90°.

Known angle
35°
Complement
55°
Sum
90°
Step 1: Identify the relationship

Complementary angles sum to 90°

Step 2: Set up the equation

35° + ? = 90°

Step 3: Solve for the unknown

? = 90° - 35° = 55°

Step 4: Verify the answer

35° + 55° = 90° ✓

The other angle measures 55°
Final answer:

The other angle measures 55°

Applied rules:

• Complementary Sum: Complementary angles always sum to 90°

• Subtraction: Missing angle = 90° - known angle

• Verification: Check that both angles add up to 90°

3 Supplementary angles
Exercise 3
Two angles are supplementary. One angle measures 110°. What is the measure of the other angle?
Definition:

Supplementary Angles: Two angles whose sum equals 180°.

Known angle
110°
Supplement
70°
Sum
180°
Step 1: Identify the relationship

Supplementary angles sum to 180°

Step 2: Set up the equation

110° + ? = 180°

Step 3: Solve for the unknown

? = 180° - 110° = 70°

Step 4: Verify the answer

110° + 70° = 180° ✓

The other angle measures 70°
Final answer:

The other angle measures 70°

Applied rules:

• Supplementary Sum: Supplementary angles always sum to 180°

• Subtraction: Missing angle = 180° - known angle

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

Key Concepts and Methods
Triangle: ∠A + ∠B + ∠C = 180°
Triangle Angle Sum
Complementary
∠A + ∠B = 90°
Right angles
Supplementary
∠A + ∠B = 180°
Straight line
Vertical Angles
∠A = ∠B
Equal angles
Key definitions:

Angle: The measure of rotation between two intersecting lines, measured in degrees (°)

Complementary Angles: Two angles that add up to 90°

Supplementary Angles: Two angles that add up to 180°

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

Vertical Angles: Opposite angles formed when two lines intersect, always equal

Complete methodology:
  1. Read the problem carefully and identify what is being asked
  2. Identify the type of angle relationship (complementary, supplementary, triangle sum, etc.)
  3. Set up an equation using the known information
  4. Solve the equation for the unknown angle
  5. Verify your answer by checking if it satisfies the angle relationship
Tip 1: Remember: Complementary = 90°, Supplementary = 180°, Triangle = 180°
Tip 2: Draw a picture to visualize the angle relationships
Tip 3: Always check your work by adding the angles back together
Tip 4: Look for keywords like "right angle", "straight line", "triangle" to identify relationships
Common errors: Mixing up complementary and supplementary, forgetting triangle sum theorem, miscalculating differences.
Real-world applications: Architecture, engineering, navigation, sports (like golf and pool shots).
Important rules to remember:

• Triangle Sum: ∠A + ∠B + ∠C = 180° for any triangle

• Complementary: Two angles sum to 90°

• Supplementary: Two angles sum to 180°

• Vertical Angles: Opposite angles are equal

Solution: Exercises 4 to 5
4 Real-world angle problem
Exercise 4
Sarah is cutting a triangular piece of fabric. She knows two angles measure 45° each. What is the measure of the third angle? What kind of triangle is this?
Definition:

Isosceles Triangle: A triangle with at least two equal angles and sides.

Known angles
45° + 45°
Sum
90°
Third angle
90°
Step 1: Add the known angles

45° + 45° = 90°

Step 2: Subtract from 180°

180° - 90° = 90°

Step 3: Identify the triangle type

Has two equal angles (45° each) and one right angle (90°)

Step 4: Draw conclusion

This is a right isosceles triangle

Third angle = 90°, Right isosceles triangle
Final answer:

The third angle measures 90°, making this a right isosceles triangle.

Applied rules:

• Triangle Sum Theorem: All triangles sum to 180°

• Isosceles Property: Equal angles correspond to equal sides

• Right Triangle: Contains one 90° angle

5 Vertical angles problem
Exercise 5
Two lines intersect forming four angles. One angle measures 70°. What are the measures of the other three angles?
Definition:

Vertical Angles: Angles opposite each other when two lines intersect, always equal.

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

Vertical angles are equal, so the angle opposite to 70° is also 70°

Step 2: Identify adjacent angles

Adjacent angles are supplementary (sum to 180°)

Step 3: Calculate adjacent angles

180° - 70° = 110°

Step 4: List all angles

70°, 110°, 70°, 110° (alternating around the intersection)

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

The four angles measure 70°, 110°, 70°, and 110°.

Applied rules:

• Vertical Angles: Opposite angles are equal

• Linear Pair: Adjacent angles sum to 180°

• Four angles: Intersecting lines form 2 pairs of equal vertical angles

Complete Guide to Angle Word Problems
Complementary: ∠A + ∠B = 90°
Complementary Angles
Key definitions:

Angle: The measure of rotation between two rays sharing a common endpoint, measured in degrees (°)

Acute Angle: An angle less than 90°

Right Angle: An angle exactly 90°

Obtuse Angle: An angle greater than 90° but less than 180°

Straight Angle: An angle exactly 180°

Complete methodology:
  1. Read the problem thoroughly and underline key information
  2. Draw a diagram to represent the situation
  3. Identify the angle relationship (complementary, supplementary, triangle sum, etc.)
  4. Set up an equation using the angle relationship
  5. Solve the equation for the unknown angle
  6. Verify your answer by checking if it makes sense in the context
Tip 1: Remember the "C" in complementary comes before "S" in supplementary, just like 90° comes before 180°.
Tip 2: Use a protractor to check your answers when possible.
Tip 3: In any triangle, if you know two angles, you can always find the third.
Tip 4: When lines intersect, they form two pairs of equal vertical angles.
Real-world applications: Architects use angles to design buildings, engineers use them for bridges, carpenters for precise cuts.
Historical context: Ancient civilizations used angle measurements for construction of pyramids, temples, and astronomical observations.
Essential formulas and rules:

• Triangle Sum: ∠A + ∠B + ∠C = 180°

• Complementary: ∠A + ∠B = 90°

• Supplementary: ∠A + ∠B = 180°

• Vertical Angles: ∠A = ∠C and ∠B = ∠D when lines intersect

• Linear Pair: Adjacent angles on a straight line sum to 180°

Visualization: Angle Relationships
Exercise 6: Angle Classification
Visualizing different angle types and their relationships:
Acute: < 90°, Right: = 90°, Obtuse: > 90° but < 180°, Straight: = 180°

Analysis: The chart shows how angles are classified by their measurements and relationships.

  • Acute angles: Less than 90° (shaded in green)
  • Right angles: Exactly 90° (marked with square corner)
  • Obtuse angles: Greater than 90° but less than 180° (shaded in blue)
  • Relationships: Complementary (sum to 90°), Supplementary (sum to 180°)

Questions & Answers

Question: I always mix up complementary and supplementary angles. How can I remember which is which?

Answer: Here are some memory tricks to help you remember:

  • Complementary = Corner (90°): Think of the "C" in Complementary for "Corner" - a right angle forms a corner
  • Supplementary = Straight (180°): Think of the "S" in Supplementary for "Straight" - a straight line is 180°
  • Alphabet order: C comes before S in the alphabet, just like 90° comes before 180°
  • Visual: A complementary angle pair looks like a right angle split in two, while a supplementary pair looks like a straight line split in two

Practice with examples: 30° and 60° are complementary (30+60=90), while 120° and 60° are supplementary (120+60=180).

Question: Do all triangles have angles that add up to 180°, even really big ones or really small ones?

Answer: Yes, absolutely! This is a fundamental law of Euclidean geometry called the Triangle Sum Theorem. It doesn't matter how big or small the triangle is, or whether it's wide and flat or tall and narrow - the three angles will always add up to 180°.

  • Even a tiny triangle drawn on paper: 30° + 60° + 90° = 180°
  • A huge triangle drawn on a playground: still adds to 180°
  • A triangle with one very large angle: like 150° + 15° + 15° = 180°

This is true because of the way parallel lines and transversals work in flat (Euclidean) space. It's one of the most important facts about triangles!

Question: How do angle relationships apply in real life? When would my child use this knowledge?

Answer: Angle relationships are used everywhere in real life! Here are practical applications:

  • Architecture: Buildings must have stable angles - right angles for walls, specific roof angles
  • Carpentry: Cutting wood at precise angles for furniture, picture frames, and construction
  • Navigation: Pilots and sailors use angles to determine direction and location
  • Sports: Pool players use angle reflection, golfers consider club angles
  • Art and Design: Creating perspective and proportion in drawings
  • Technology: Cameras, telescopes, and microscopes use lens angles

Understanding angles helps children develop spatial reasoning skills that are valuable in many careers and everyday problem-solving situations.