Solved Exercises on Types of Angle Pairs in Grade 4

Master types of angle pairs: complementary, supplementary, vertical, adjacent, and linear pairs through these 5 detailed exercises with visual diagrams.

Solution: Exercises 1 to 3
1 Complementary Angles
Exercise 1
Two angles are complementary. If one angle measures 35°, what is the measure of the other angle?
Definition:

Complementary angles: Two angles whose sum equals 90°

Method:
  1. Identify that the angles are complementary (sum = 90°)
  2. Set up the equation: Angle 1 + Angle 2 = 90°
  3. Solve for the unknown angle
35°
?
Given
35° + ? = 90°
Unknown
? = 90° - 35°
Result
55°
Step 1: Apply complementary angle rule

Complementary angles sum to 90°: 35° + ? = 90°

Step 2: Solve for the unknown angle

? = 90° - 35° = 55°

Step 3: Verify the answer

35° + 55° = 90° ✓

The other angle measures 55°
Final answer:

The other angle measures 55°

Applied rules:

• Complementary angles: Sum equals 90°

• Subtraction: To find one angle, subtract the known angle from 90°

• Verification: Check that the sum equals 90°

2 Supplementary Angles
Exercise 2
Two angles are supplementary. If one angle measures 120°, what is the measure of the other angle?
Definition:

Supplementary angles: Two angles whose sum equals 180°

120°
?
Expression
120° + ? = 180°
Solution
? = 180° - 120°
Result
60°
Step 1: Apply supplementary angle rule

Supplementary angles sum to 180°: 120° + ? = 180°

Step 2: Solve for the unknown angle

? = 180° - 120° = 60°

Step 3: Verify the answer

120° + 60° = 180° ✓

The other angle measures 60°
Final answer:

The other angle measures 60°

Applied rules:

• Supplementary angles: Sum equals 180°

• Subtraction: To find one angle, subtract the known angle from 180°

• Verification: Check that the sum equals 180°

3 Vertical Angles
Exercise 3
Two lines intersect forming four angles. One angle measures 75°. What are the measures of the other three angles?
Definition:

Vertical angles: Angles opposite each other when two lines intersect. They are equal.

75°
75°
105°
105°
Given
One angle = 75°
Vertical angle
= 75°
Adjacent angles
= 180° - 75° = 105°
Step 1: Identify vertical angles

Vertical angles are equal, so one angle is also 75°

Step 2: Find adjacent angles

Adjacent angles are supplementary: 180° - 75° = 105°

Step 3: Identify the fourth angle

The fourth angle is vertical to the 105° angle, so it's also 105°

The four angles are: 75°, 105°, 75°, 105°
Final answer:

The other three angles measure 75°, 105°, and 105°

Applied rules:

• Vertical angles: Opposite angles are equal

• Linear pairs: Adjacent angles sum to 180°

• Four angles: Total sum around a point is 360°

Angle Pair Types and Properties
∠A + ∠B = 90°
Complementary Angles
Complementary
∠A + ∠B = 90°
Angles that sum to 90°
Supplementary
∠A + ∠B = 180°
Angles that sum to 180°
Vertical
∠A = ∠B
Opposite angles formed by intersecting lines
Adjacent angles: Share a common vertex and side, but no common interior points.
Linear pair: Adjacent angles that are supplementary (form a straight line).
Vertical angles: Always equal when two lines intersect.
Solution: Exercises 4 to 5
4 Linear Pairs
Exercise 4
Two angles form a linear pair. If one angle is twice as large as the other, find the measures of both angles.
Definition:

Linear pair: Two adjacent angles that form a straight line (sum = 180°)

x°
2x°
Setup
x + 2x = 180°
Combine
3x = 180°
Solve
x = 60°
Step 1: Set up the equation

Linear pairs sum to 180°: x + 2x = 180°

Step 2: Combine like terms

3x = 180°

Step 3: Solve for x

x = 180° ÷ 3 = 60°

Step 4: Find both angles

First angle: x = 60°, Second angle: 2x = 120°

The angles are 60° and 120°
Final answer:

The angles measure 60° and 120°

Applied rules:

• Linear pairs: Sum equals 180°

• Algebra: Set up equations based on given relationships

• Verification: 60° + 120° = 180° ✓

5 Adjacent Angles
Exercise 5
Two adjacent angles form an angle of 150°. If one angle is 30° more than the other, find the measures of both angles.
Definition:

Adjacent angles: Angles that share a common vertex and side, but no common interior points

x°
(x+30)°
Setup
x + (x + 30) = 150°
Combine
2x + 30 = 150°
Solve
x = 60°
Step 1: Set up the equation

Adjacent angles sum to total: x + (x + 30) = 150°

Step 2: Simplify the equation

2x + 30 = 150°

Step 3: Solve for x

2x = 120°, so x = 60°

Step 4: Find both angles

First angle: x = 60°, Second angle: x + 30 = 90°

Step 5: Verify the result

60° + 90° = 150° ✓

The angles are 60° and 90°
Final answer:

The angles measure 60° and 90°

Applied rules:

• Adjacent angles: Share vertex and side

• Algebra: Express relationships with variables

• Verification: Always check the sum

Key Concepts and Properties
∠A + ∠B = 180°
Linear Pair
Key definitions:

Complementary angles: Two angles that sum to 90°

Supplementary angles: Two angles that sum to 180°

Vertical angles: Opposite angles formed by intersecting lines (equal)

Adjacent angles: Angles that share a common vertex and side

Linear pair: Adjacent angles that are supplementary (form a straight line)

Problem-solving approach:
  1. Identify the type of angle pair: Determine if complementary, supplementary, vertical, etc.
  2. Set up an equation: Use the defining relationship (sum = 90°, 180°, or equality)
  3. Solve the equation: Find the unknown angle(s)
  4. Verify the answer: Check that the relationship holds true
Tip 1: Remember: C for Complementary (Corner/Cute - 90°), S for Supplementary (Straight - 180°).
Tip 2: Vertical angles are always equal - look for the X pattern when lines intersect.
Tip 3: Linear pairs always form a straight line and sum to 180°.
Tip 4: Always verify your answers by checking that the angle relationships hold.
Common errors: Confusing complementary with supplementary, forgetting that vertical angles are equal.
Memory aids: Complementary = Corner (90°), Supplementary = Straight (180°).
Essential facts to remember:

• Complementary: ∠A + ∠B = 90°

• Supplementary: ∠A + ∠B = 180°

• Vertical angles: ∠A = ∠B (when lines intersect)

• Linear pairs: Adjacent and supplementary (∠A + ∠B = 180°)

• Adjacent angles: Share vertex and side but no interior points

Visual Guide: Angle Pair Relationships
Complementary
30° + 60° = 90°
Supplementary
120° + 60° = 180°
Vertical
Equal angles across from each other
Linear Pair
Sum to 180°

Summary: Different angle pairs have distinct properties:

  • Complementary angles form corner-like shapes and sum to 90°
  • Supplementary angles form straight lines and sum to 180°
  • Vertical angles are opposite each other when lines cross and are equal
  • Linear pairs are adjacent and supplementary, forming a straight line

Questions & Answers

Question: I get confused between complementary and supplementary angles. How can I remember which is which?

Answer: Great question! Here are some memory tricks:

  • Complementary starts with C - like Corner. Think of a corner of a square, which is 90°.
  • Supplementary starts with S - like Straight. Think of a straight line, which is 180°.
  • Also, C comes before S in the alphabet, just like 90° comes before 180° in number order!

So remember: Complementary = Corner (90°) and Supplementary = Straight (180°).

Practice by looking around for 90° corners (complementary) and 180° straight lines (supplementary) to reinforce the concepts!

Question: When two lines cross, why are the opposite angles always equal?

Answer: These opposite angles are called vertical angles, and they're always equal due to a mathematical property!

Think about it this way: when two lines intersect, they form four angles around the point where they meet. The angles next to each other (adjacent) are supplementary (they add up to 180°).

If angle A and angle B are adjacent and add to 180°, and angle A and angle C (across from angle B) are also adjacent and add to 180°, then angle B must equal angle C!

This is because both angle B and angle C are what's needed to make angle A equal 180°. Since they're both "what's needed," they must be equal!

It's like if you and your friend both need the same amount of money to buy the same toy, you both have the same amount!

Question: How do I know if angles are adjacent? Sometimes I think they are, but I'm wrong.

Answer: Adjacent angles have to meet three specific requirements:

  1. Share the same vertex: They have the same corner point
  2. Share a common side: They share one ray (the line that goes out from the corner)
  3. No overlapping interiors: They don't overlap inside - there's no shared area between them

Think of it like two slices of pizza that share the same center point and one edge of crust, but don't overlap on the pizza surface.

A good way to check: Put your finger on the vertex (corner), trace along the shared side, and see if you can go to both angles without lifting your finger from the shared part.

If you can, they might be adjacent. If there's any overlap in the "inside" parts of the angles, they're not adjacent.

Question: What's the difference between a linear pair and just supplementary angles?

Answer: Great observation! There is a key difference:

  • Supplementary angles: Any two angles that add up to 180°, regardless of their position
  • Linear pair: Two angles that are BOTH adjacent AND supplementary (they form a straight line)

So all linear pairs are supplementary angles, but not all supplementary angles are linear pairs!

Think of it this way: A linear pair is a special type of supplementary angle pair that has the extra requirement of being adjacent (sharing a vertex and side).

Example: Two separate angles that each measure 90° are supplementary (90° + 90° = 180°), but they're not a linear pair because they're not adjacent. However, two angles that form a straight line are both supplementary AND a linear pair.

Question: How can I check if my answers are correct when solving angle pair problems?

Answer: Here are several ways to verify your angle pair solutions:

  1. Check the defining relationship:
    • For complementary: Do the angles add up to 90°?
    • For supplementary: Do the angles add up to 180°?
    • For vertical: Are opposite angles equal?
  2. Draw a diagram: Sketch the situation to visually confirm your answer makes sense
  3. Use logical reasoning: Does your answer fit the context of the problem?
  4. Plug back in: Substitute your answer back into the original conditions

For example, if you found that two complementary angles are 35° and 55°, check: 35° + 55° = 90° ✓

Always take a moment to verify your work - it's much easier to catch small mistakes early than to fix them later!