Solved Exercises on Angles and Lines at a Point in Grade 4

Master angles and lines at a point: finding missing angles, calculating angles around a point, and understanding angle relationships through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Finding angles around a point
Exercise 1
Four angles meet at a point. Three of the angles measure 80°, 70°, and 90°. What is the measure of the fourth angle?
Definition:

Angles at a point: The sum of all angles around a point is always 360°

Solution method:
  1. Add the known angles together
  2. Subtract the sum from 360°
  3. The result is the missing angle
Known angles
80° + 70° + 90°
Total
240°
Missing angle
360° - 240° = 120°
Step 1: Add the known angles

80° + 70° + 90° = 240°

Step 2: Apply the rule for angles at a point

Sum of all angles around a point = 360°

Step 3: Calculate the missing angle

Missing angle = 360° - 240° = 120°

Step 4: Verify the answer

80° + 70° + 90° + 120° = 360° ✓

The fourth angle measures 120°
Final answer:

The measure of the fourth angle is 120°.

Applied rules:

• Angles at a point: Sum to 360°

• Subtraction: To find the missing part

• Verification: Check that all angles sum to 360°

2 Angles on a straight line
Exercise 2
Two angles form a straight line. One angle measures 130°. What is the measure of the other angle?
Definition:

Angles on a straight line: Two angles that form a straight line are supplementary and sum to 180°

Known angle
130°
Straight line
180°
Missing angle
50°
Step 1: Understand the relationship

Two angles forming a straight line are supplementary, so they add up to 180°

Step 2: Set up equation

Known angle + Unknown angle = 180°

130° + Unknown angle = 180°

Step 3: Solve for unknown angle

Unknown angle = 180° - 130° = 50°

Step 4: Verify the answer

130° + 50° = 180° ✓

The missing angle measures 50°
Final answer:

The measure of the other angle is 50°.

Applied rules:

• Supplementary angles: Sum to 180°

• Equation solving: Subtract known angle from total

• Verification: Check that angles sum to 180°

3 Opposite angles at a point
Exercise 3
Two lines intersect at point O. One of the angles formed measures 55°. What are the measures of the other three angles?
Definition:

Opposite angles: When two lines intersect, opposite angles are equal (vertical angles)

Given angle
55°
Opposite angle
55°
Adjacent angles
125°
Step 1: Identify opposite angles

When two lines intersect, the angles opposite each other are equal. So one of the other angles is also 55°.

Step 2: Find adjacent angles

Adjacent angles form a straight line, so they are supplementary (add up to 180°).

Adjacent angle = 180° - 55° = 125°

Step 3: Identify all four angles

There are two angles of 55° (opposite each other) and two angles of 125° (opposite each other).

Step 4: Verify the answer

55° + 125° + 55° + 125° = 360° ✓

The four angles are 55°, 125°, 55°, and 125°
Final answer:

The measures of the four angles are 55°, 125°, 55°, and 125°.

Applied rules:

• Vertical angles: Opposite angles are equal

• Supplementary angles: Adjacent angles sum to 180°

• Angles at a point: All four angles sum to 360°

Rules and methods, laws,...
Angles at a point = 360°
Key Rule
Angles at a Point
Sum = 360°
All angles around a point
Angles on a Line
Sum = 180°
Supplementary angles
Vertical Angles
Equal
Opposite angles are equal

Angle Relationships

📐
Angles at a Point
When multiple angles meet at a single point, they form a full circle. The sum of all angles around a point is always 360°.
Example: A + B + C + D = 360°
Angles on a Line
When two angles form a straight line, they are supplementary and always add up to 180°.
Example: A + B = 180°
Vertical Angles
When two lines intersect, the angles opposite each other (vertical angles) are always equal.
Example: A = C, B = D
Problem-Solving Steps
1
Identify Type
2
Apply Rule
3
Calculate
4
Verify
Key Concepts & Practice Tips
A + B + C + D = 360°
Angles at a Point
Key definitions:

Point: A specific location where lines or rays meet

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

Line: A straight path that extends infinitely in both directions

Angle: The space between two rays that share a common endpoint (vertex)

Complete methodology:
  1. Identify the situation: Determine if angles are at a point, on a line, or vertical angles
  2. Select the rule: Apply the appropriate rule (360° for point, 180° for line, equality for vertical)
  3. Set up equation: Write an equation using the known and unknown angles
  4. Solve: Calculate the unknown angle(s)
  5. Verify: Check that your answer makes sense in the context
Tip 1: Think of angles at a point as slices of a pie that add up to a whole pie (360°).
Tip 2: When two lines cross, the angles across from each other are always equal.
Tip 3: Always check your answer by adding all angles to ensure they sum correctly.
Tip 4: If you know one angle in a pair of vertical angles, you automatically know the other.
Common errors: Forgetting that angles at a point sum to 360°, confusing vertical angles with adjacent angles.
Exam preparation: Practice identifying angle types, memorize key rules, work on verification skills.
Facts to remember:

• Angles at a point sum to 360°

• Angles on a straight line sum to 180°

• Vertical angles are equal

• Adjacent angles on a line are supplementary

• All angles around a point form a complete circle

Comprehensive Summary: Angles and Lines at a Point

This summary covers all essential concepts, rules, and methods for understanding angles and lines at a point.

Definitions

Point: A specific location in space where lines or rays meet.

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

Line: A straight path that extends infinitely in both directions.

Angle: The space between two rays that share a common endpoint (vertex).

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

Core Rules

Angles at a point: The sum of all angles around a point is always 360°.

Angles on a straight line: Two angles that form a straight line sum to 180° (supplementary).

Vertical angles: When two lines intersect, opposite angles are equal.

Adjacent angles: Angles that share a common side and vertex.

Methods & Procedures

Finding missing angles: Add known angles and subtract from total (360° or 180°).

Identifying angle types: Look for intersecting lines, straight lines, or angles around a point.

Verification: Always check that angles sum to the expected total.

Examples

Simple: Two angles on a line: 60° + ? = 180° → ? = 120°.

Intermediate: Three angles at a point: 100° + 80° + ? = 360° → ? = 180°.

Advanced: Intersecting lines with one angle 40° → vertical = 40°, adjacent = 140°.

Tips & Tricks

Memory aid: "At a point" = "A full circle" = "360°".

Visualization: Imagine a compass or clock face for angles at a point.

Check work: Add all angles to verify they equal expected sum.

Common Mistakes

Confusing totals: Using 180° instead of 360° for angles at a point.

Misidentifying angles: Thinking adjacent angles are equal instead of supplementary.

Calculation errors: Making arithmetic mistakes when solving for missing angles.

Questions & Answers

Question: Why do angles at a point always add up to 360°? How can I remember this?

Answer: This happens because a full rotation around a point creates a complete circle, and a circle is defined as having 360°. Think of it this way:

  • If you start facing North and turn completely around to face North again, you've made a full rotation of 360°.
  • Similarly, if you walk around the edge of a circular track, you travel 360° to return to your starting point.

To remember this: Think of a clock face - if you start at 12 o'clock and go all the way around back to 12 o'clock, you've completed a full circle of 360°.

You can also visualize a compass - North, East, South, West form four 90° angles that sum to 360°.

Question: When two lines cross, why are the angles across from each other equal? It doesn't look like they should be!

Answer: These are called "vertical angles," and they are always equal due to mathematical properties:

  • When two lines intersect, they form four angles.
  • Each pair of adjacent angles forms a straight line (180°).
  • If one angle is x°, then its adjacent angle must be (180° - x°).
  • The angle opposite to x° shares the same adjacent angle (180° - x°), so it must also be x°.

Even though they might not look equal visually, the mathematical relationship ensures they are always the same size.

This is a fundamental geometric property that always holds true!

Question: How can I quickly check if my answer is correct when solving for missing angles?

Answer: Here are quick verification methods:

  • Angles at a point: Add all angles together - they should equal 360°.
  • Angles on a straight line: Two angles should add up to 180°.
  • Vertical angles: Make sure the opposite angles are equal.
  • Reasonableness check: Does your answer make sense? (e.g., an acute angle shouldn't be 150°)

You can also substitute your answer back into the original problem to ensure all conditions are met.

Developing this verification habit will help you catch errors and build confidence in your solutions!