Solved Exercises on Angles and Angle Relationships in Grade 8

Master angles and angle relationships: complementary, supplementary, vertical, adjacent, and transversal angles through these 5 detailed exercises.

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

Complementary angles: Two angles whose sum equals 90°

Solution method:
  1. Identify the relationship: complementary angles sum to 90°
  2. Set up the equation: angle₁ + angle₂ = 90°
  3. Solve for the unknown angle
Given
One angle = 35°
Equation
35° + x = 90°
Solution
x = 55°
Step 1: Understand the definition

Complementary angles add up to 90°

Step 2: Set up the equation

Known angle + Unknown angle = 90°

35° + x = 90°

Step 3: Solve for the unknown

x = 90° - 35° = 55°

The other angle measures 55°
Final answer:

The other angle measures 55°

Applied rules:

• Definition: Complementary angles sum to 90°

• Algebra: Isolate the unknown variable

• Verification: Check that 35° + 55° = 90° ✓

2 Supplementary Angles
Exercise 2
Two angles are supplementary. One angle is twice the measure of the other. Find the measures of both angles.
Definition:

Supplementary angles: Two angles whose sum equals 180°

Given
x + 2x = 180°
Simplify
3x = 180°
Solution
x = 60°, 2x = 120°
Step 1: Define variables

Let the smaller angle be x°

Then the larger angle is 2x°

Step 2: Apply supplementary rule

Sum of supplementary angles = 180°

x + 2x = 180°

Step 3: Solve the equation

3x = 180°

x = 60°

So 2x = 120°

The angles measure 60° and 120°
Final answer:

The angles measure 60° and 120°

Applied rules:

• Definition: Supplementary angles sum to 180°

• Algebra: Set up and solve linear equations

• Verification: Check that 60° + 120° = 180° ✓

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

Vertical angles: Opposite angles formed by intersecting lines, which are equal

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

Vertical angles are equal, so the opposite angle is also 75°

Step 2: Apply supplementary angles rule

Adjacent angles form a straight line, so they sum to 180°

Adjacent angle = 180° - 75° = 105°

Step 3: Find all four angles

Two angles measure 75° (vertical angles)

Two angles measure 105° (vertical angles)

Step 4: Verify the solution

75° + 105° + 75° + 105° = 360° ✓

The four angles measure 75°, 105°, 75°, and 105°
Final answer:

The four angles measure 75°, 105°, 75°, and 105°

Applied rules:

• Vertical angles theorem: Vertical angles are congruent

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

• Complete rotation: All angles around a point sum to 360°

Rules and methods, laws,...
∠A + ∠B = 90°
Complementary Angles
∠A + ∠B = 180°
Supplementary Angles
∠A = ∠B
Vertical Angles
Adjacent Angles
∠A + ∠B = Linear Pair
Share a common vertex and side
Linear Pair
∠A + ∠B = 180°
Adjacent angles forming a straight line
Transversal Angles
Corresponding ≅, Alternate ≅
Angles formed by parallel lines cut by a transversal
Key definitions:

Angle: Figure formed by two rays sharing a common endpoint

Complementary: Two angles that sum to 90°

Supplementary: Two angles that sum to 180°

Vertical: Opposite angles formed by intersecting lines

Adjacent: Angles sharing a common vertex and side

Complete methodology:
  1. Identify the angle relationship: Complementary, supplementary, vertical, etc.
  2. Set up the equation: Based on the relationship rule
  3. Solve for the unknown: Using algebraic methods
  4. Verify the solution: Check against the original conditions
Tip 1: Remember "C" for Complementary means 90° (Corner angle).
Tip 2: Remember "S" for Supplementary means 180° (Straight line).
Tip 3: Vertical angles are always equal and form an "X".
Tip 4: Adjacent angles share a vertex and a side.
Common errors: Confusing complementary with supplementary, forgetting vertical angles are equal, misidentifying adjacent angles.
Exam preparation: Memorize angle sum relationships, practice identifying angle pairs, work on algebraic setup for unknowns.
Formulas to know by heart:

• Complementary: ∠A + ∠B = 90°

• Supplementary: ∠A + ∠B = 180°

• Vertical angles: ∠A = ∠B

• Linear pair: ∠A + ∠B = 180°

• Corresponding angles (parallel lines): Equal

• Alternate interior angles (parallel lines): Equal

Solution: Exercises 4 to 5
4 Transversal and Parallel Lines
Exercise 4
A transversal cuts two parallel lines. One angle formed is 110°. Find the measures of all eight angles formed.
Definition:

Transversal: A line that intersects two or more parallel lines

Given
One angle = 110°
Vertical angles
= 110°
Supplementary
= 70°
Step 1: Identify the given angle

Let's say angle 1 = 110°

Step 2: Find vertical angle

Angle 3 (vertical to angle 1) = 110°

Step 3: Find supplementary angles

Angle 2 and angle 4 are supplementary to angle 1

Angle 2 = Angle 4 = 180° - 110° = 70°

Step 4: Apply parallel line theorems

Corresponding angles are equal:

Angle 5 = Angle 1 = 110°

Angle 6 = Angle 2 = 70°

Angle 7 = Angle 3 = 110°

Angle 8 = Angle 4 = 70°

Four angles = 110°, Four angles = 70°
Final answer:

Four angles measure 110° and four angles measure 70°

Applied rules:

• Vertical angles: Opposite angles are equal

• Linear pairs: Adjacent angles sum to 180°

• Parallel line theorems: Corresponding angles are equal

• Parallel line theorems: Alternate interior angles are equal

5 Adjacent and Linear Pairs
Exercise 5
Two adjacent angles form a linear pair. The ratio of their measures is 2:3. Find the measures of both angles.
Definition:

Linear pair: Two adjacent angles whose non-common sides form a straight line

Ratio
2x : 3x
Linear pair
2x + 3x = 180°
Solution
x = 36°
Step 1: Set up the ratio

If the ratio is 2:3, then the angles are 2x and 3x

Step 2: Apply linear pair rule

Linear pairs sum to 180°

2x + 3x = 180°

Step 3: Solve for x

5x = 180°

x = 36°

Step 4: Find both angles

First angle = 2x = 2(36°) = 72°

Second angle = 3x = 3(36°) = 108°

Step 5: Verify the solution

72° + 108° = 180° ✓

Ratio 72:108 = 2:3 ✓

The angles measure 72° and 108°
Final answer:

The angles measure 72° and 108°

Applied rules:

• Linear pair: Adjacent angles forming a straight line sum to 180°

• Ratio: Express quantities in terms of a common variable

• Algebra: Solve equations with multiple variables

Key Concepts, Laws, Methods, and Formulas for Angles
∠A + ∠B = 90°
Complementary Angles
Key definitions:

Angle: A figure formed by two rays sharing a common endpoint called the vertex

Acute angle: An angle measuring less than 90°

Right angle: An angle measuring exactly 90°

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

Straight angle: An angle measuring exactly 180°

Reflex angle: An angle measuring greater than 180° but less than 360°

Complementary angles: Two angles whose sum is 90°

Supplementary angles: Two angles whose sum is 180°

Vertical angles: Opposite angles formed by intersecting lines, which are always equal

Adjacent angles: Two angles that share a common vertex and side but do not overlap

Linear pair: Two adjacent angles whose non-common sides form a straight line, always supplementary

Complete methodology:
  1. Identify the angle relationship: Determine if angles are complementary, supplementary, vertical, adjacent, or part of parallel lines cut by a transversal
  2. Recall the relevant theorem or property: Apply the appropriate angle relationship rule
  3. Set up the equation: Based on the relationship, create an equation involving the angle measures
  4. Solve for the unknown: Use algebraic methods to find missing angle measures
  5. Verify the solution: Check that the solution satisfies all conditions of the problem
Tip 1: Remember C-90 (Complementary = 90°) and S-180 (Supplementary = 180°).
Tip 2: Vertical angles form an "X" and are always equal.
Tip 3: When parallel lines are cut by a transversal, look for F-patterns (corresponding) and Z-patterns (alternate).
Tip 4: Linear pairs are always supplementary (add to 180°).
Tip 5: Adjacent angles share a common side and vertex but do not overlap.
Common errors: Mixing up complementary and supplementary, forgetting vertical angles are equal, misidentifying angle pairs in complex diagrams.
Memory aids: Complementary = Corner (90°), Supplementary = Straight line (180°), Vertical = Opposite (equal).
Problem-solving strategies: Label all known angles, use geometric properties to find unknowns, check that your answer makes sense in the context of the diagram.
Essential formulas and theorems:

• Complementary angles: ∠A + ∠B = 90°

• Supplementary angles: ∠A + ∠B = 180°

• Vertical angles: ∠A = ∠B (opposite angles formed by intersecting lines)

• Linear pair: ∠A + ∠B = 180° (adjacent angles forming a straight line)

• Corresponding angles (parallel lines): Equal

• Alternate interior angles (parallel lines): Equal

• Alternate exterior angles (parallel lines): Equal

• Co-interior angles (parallel lines): Supplementary

• Sum of angles around a point: 360°

• Sum of angles on a straight line: 180°

Visual Representation: Angle Relationships
Exercise 6: Understanding Angle Types
Visual representation of different angle relationships:
- Complementary angles (sum to 90°)
- Supplementary angles (sum to 180°)
- Vertical angles (opposite and equal)
- Adjacent angles (share a vertex and side)

Analysis: The chart illustrates how different angle relationships work in geometric figures.

  • Complementary angles form a corner (90°)
  • Supplementary angles form a straight line (180°)
  • Vertical angles are opposite each other when lines intersect
  • Adjacent angles share a common ray

Questions & Answers

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

Answer: Great question! Here are some memory tricks:

  • Complementary angles make a Corner (90°): Think of the "C" in complementary looking like a corner of a square.
  • Supplementary angles make a Straight line (180°): Think of the "S" in supplementary looking like a straight line.
  • Picture a corner of a room (90°) for complementary.
  • Picture a flat surface or straight angle (180°) for supplementary.

Also, remember that "co-" means together, and together they make a corner (90°). "Sup-" means added to, and when added together they make a straight line (180°).

Practice with simple examples: 30° and 60° are complementary because 30 + 60 = 90°. 110° and 70° are supplementary because 110 + 70 = 180°.

Question: When two lines intersect, how many pairs of vertical angles are formed and why are they equal?

Answer: When two lines intersect, they form 4 angles total, which create 2 pairs of vertical angles.

  • Vertical angles are opposite each other across the intersection point
  • They are equal because they are formed by the same two intersecting lines
  • This can be proven using the fact that adjacent angles form linear pairs (sum to 180°)

Proof: If angle A and angle B are adjacent and form a linear pair, then A + B = 180°. If angle B and angle C are adjacent and form a linear pair, then B + C = 180°. Therefore, A = C (since both equal 180° - B).

This relationship is fundamental in geometry and helps solve many angle problems quickly.

Question: What's the difference between adjacent angles and a linear pair? Aren't they the same thing?

Answer: Great question! They're related but not the same:

  • Adjacent angles: Two angles that share a common vertex and a common side, but don't overlap
  • Linear pair: A specific type of adjacent angles where the non-common sides form a straight line
  • All linear pairs are adjacent angles, but not all adjacent angles are linear pairs

Key differences:

  • Adjacent angles just need to share a vertex and side
  • Linear pairs must additionally form a straight line (sum to 180°)
  • Linear pairs are always supplementary, adjacent angles may or may not be

Think of adjacent angles as any two angles that are next to each other, while linear pairs are specifically those adjacent angles that create a straight line.

Question: When solving problems with parallel lines and transversals, how do I identify corresponding angles?

Answer: Corresponding angles are easy to identify once you know the pattern:

  • Look for the "F" shape in the diagram (the transversal creates the crossbar of the F)
  • Corresponding angles are in matching positions relative to the parallel lines and transversal
  • They are on the same side of the transversal and in the same position relative to the parallel lines

For example, if you have two parallel lines with a transversal cutting through them:

  • Upper left angle on first line corresponds to upper left angle on second line
  • Lower right angle on first line corresponds to lower right angle on second line
  • And so on for all four pairs

Remember: Corresponding angles are always equal when lines are parallel. This is one of the fundamental theorems of parallel lines.

Question: How can I check if my angle calculations are correct in complex diagrams?

Answer: Here are several verification techniques:

  1. Check angle sums: Complementary angles should sum to 90°, supplementary to 180°, angles around a point to 360°
  2. Use multiple relationships: Verify using a different angle relationship than the one you originally used
  3. Substitute back: Plug your answer into the original equation to see if it balances
  4. Visual estimation: Does your answer match what you'd expect from looking at the diagram?
  5. Pattern checking: In parallel line problems, verify that corresponding angles are equal, alternate angles are equal, etc.

Always write down which theorem or property you're using for each calculation. This makes it easier to trace back if you need to verify your work.

Final tip: Draw a small sketch of just the angles you're working with if the full diagram is too complex.