Solved Exercises on Angles and Angle Relationships in Grade 7

Master angles and angle relationships: complementary, supplementary, vertical, adjacent, and linear pairs 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 is 90°. They form a right angle when placed together.

Method for complementary angles:
  1. Recall that complementary angles sum to 90°
  2. Set up the equation: Angle₁ + Angle₂ = 90°
  3. Solve for the unknown angle
  4. Verify the solution
Given
One angle = 35°
Formula
A + B = 90°
Find
Other angle = 55°
Step 1: Recall complementary angle definition

Complementary angles sum to 90°

Step 2: Set up the equation

Angle₁ + Angle₂ = 90°

35° + Angle₂ = 90°

Step 3: Solve for the unknown angle

Angle₂ = 90° - 35°

Angle₂ = 55°

Step 4: Verify the solution

35° + 55° = 90° ✓

The other angle measures 55°
Final answer:

The measure of the other angle is 55°.

Applied rules:

• Complementary Sum: A + B = 90°

• Algebraic Solution: Isolate the unknown variable

• Verification: Check that angles sum to 90°

2 Supplementary Angles
Exercise 2
Two angles are supplementary. One angle is 4 times larger than the other. Find the measures of both angles.
Definition:

Supplementary Angles: Two angles whose sum is 180°. They form a straight line when placed together.

Relationship
A + B = 180°
Given
A = 4B
Solution
36°, 144°
Step 1: Define variables

Let the smaller angle = x°

Then the larger angle = 4x°

Step 2: Set up the equation

Since angles are supplementary:

x + 4x = 180°

Step 3: Solve the equation

5x = 180°

x = 36°

4x = 4 × 36° = 144°

Step 4: Verify the solution

36° + 144° = 180° ✓

144° = 4 × 36° ✓

The angles measure 36° and 144°
Final answer:

The measures of the angles are 36° and 144°.

Applied rules:

• Supplementary Sum: A + B = 180°

• Algebraic Representation: Use variables to express relationships

• Equation Solving: Combine like terms and isolate variable

3 Vertical Angles
Exercise 3
Two lines intersect forming four angles. If one of the angles measures 70°, find the measures of the other three angles.
Definition:

Vertical Angles: Opposite angles formed when two lines intersect. They are always equal in measure.

Given
One angle = 70°
Vertical
Opposite = 70°
Adjacent
110° each
Step 1: Identify the angle relationships

When two lines intersect, they form 4 angles

Vertical angles are opposite each other and equal

Adjacent angles are supplementary (sum to 180°)

Step 2: Find the vertical angle

The angle opposite to 70° is also 70° (vertical angles are equal)

Step 3: Find the adjacent angles

Adjacent angles are supplementary to 70°

Adjacent angle = 180° - 70° = 110°

Step 4: Identify all four angles

Angle 1: 70°

Angle 2: 110° (adjacent to 70°)

Angle 3: 70° (vertical to angle 1)

Angle 4: 110° (vertical to angle 2)

Step 5: Verify the solution

70° + 110° + 70° + 110° = 360° ✓

Vertical angles are equal: 70° = 70°, 110° = 110° ✓

The angles are: 70°, 110°, 70°, 110°
Final answer:

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

Applied rules:

• Vertical Angles Theorem: Vertical angles are congruent

• Linear Pair: Adjacent angles sum to 180°

• Full Circle: Four angles around a point sum to 360°

Angle Relationships and Properties
\(\angle A + \angle B = 90° \text{ (complementary)}\)
Complementary Angles
Complementary
A + B = 90°
Right angle sum
Supplementary
A + B = 180°
Straight line sum
Vertical
A = B
Opposite angles equal
Key definitions:

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

Complementary Angles: Two angles whose measures add up to 90°.

Supplementary Angles: Two angles whose measures add up to 180°.

Vertical Angles: Angles opposite each other when two lines intersect.

Adjacent Angles: Angles that share a common vertex and side but do not overlap.

Linear Pair: Adjacent angles that form a straight line (supplementary).

Complete methodology:
  1. Identify Angle Type: Determine the relationship between angles
  2. Apply Relevant Property: Use the appropriate angle relationship rule
  3. Set Up Equation: Create an equation based on the relationship
  4. Solve Algebraically: Find the unknown angle(s)
  5. Verify Solution: Check that the answer satisfies the angle relationship
  6. Check Reasonableness: Ensure the answer makes sense geometrically
Tip 1: Remember: 'Co' for corner (90°), 'Su' for straight (180°).
Tip 2: Vertical angles are always equal - no matter what!
Tip 3: Linear pairs are always supplementary.
Tip 4: Draw a diagram to visualize the angle relationships.
Common errors: Confusing complementary and supplementary, forgetting vertical angles are equal, misidentifying adjacent angles.
Exam preparation: Practice identifying angle relationships in complex diagrams, work with algebraic expressions for angles, solve multi-step problems.
Formulas to know by heart:

• Complementary: ∠A + ∠B = 90°

• Supplementary: ∠A + ∠B = 180°

• Vertical Angles: ∠A = ∠B

• Linear Pair: ∠A + ∠B = 180°

• Angles around a point: Sum = 360°

• Angles on a straight line: Sum = 180°

Solution: Exercises 4 to 5
4 Adjacent and Linear Pair
Exercise 4
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 that form a straight line. They are always supplementary (sum to 180°).

Ratio
2:3
Linear Pair
Sum = 180°
Solution
72°, 108°
Step 1: Understand the problem

Two adjacent angles form a linear pair, so they are supplementary

Their ratio is 2:3

Step 2: Set up the ratio

Let the angles be 2x and 3x degrees

Step 3: Apply the linear pair property

Since they form a linear pair: 2x + 3x = 180°

Step 4: Solve for x

5x = 180°

x = 36°

Step 5: Find both angles

First angle: 2x = 2 × 36° = 72°

Second angle: 3x = 3 × 36° = 108°

Step 6: Verify the solution

72° + 108° = 180° ✓

Ratio: 72:108 = 2:3 ✓

The angles measure 72° and 108°
Final answer:

The measures of the angles are 72° and 108°.

Applied rules:

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

• Ratio Representation: Express angles as multiples of a variable

• Algebraic Solution: Solve equations to find unknown values

5 Complex Angle Relationships
Exercise 5
Three lines intersect at a point, forming six angles. One angle measures 50°. Find the measures of all other angles, assuming the lines form a symmetrical pattern.
Definition:

Angle Relationships at Intersection: When multiple lines intersect, vertical angles are equal, and adjacent angles on a straight line are supplementary.

Given
One angle = 50°
Symmetry
Pattern repeats
Solution
50°, 130°, 50°, 130°, 50°, 130°
Step 1: Analyze the intersection

Three lines intersecting at one point create 6 angles around the point

These angles alternate between two different measures

Step 2: Apply vertical angles theorem

Vertical angles are equal, so every other angle equals 50°

Step 3: Find supplementary angles

Adjacent angles on a straight line are supplementary

Other angle = 180° - 50° = 130°

Step 4: Identify all six angles

Starting from the given 50° angle and moving around the point:

50°, 130°, 50°, 130°, 50°, 130°

Step 5: Verify the solution

Sum of all angles around a point: 50° + 130° + 50° + 130° + 50° + 130° = 540°

Wait, this is incorrect. Let me recalculate.

Actually, three lines intersecting form 6 angles that sum to 360° around the point.

Correct: 50° + 130° + 50° + 130° + 50° + 130° = 540°

No, that's still wrong. With 3 lines, we have 6 angles alternating: 50°, 130°, 50°, 130°, 50°, 130° = 540°

This is impossible since angles around a point sum to 360°.

Let me reconsider: 3 lines through a point create 6 angles, but they're paired as 3 sets of vertical angles.

So we have 3 pairs of equal angles. If one is 50°, its vertical is 50°.

The adjacent angles are 130° each (since 50° + 130° = 180°).

Actually, 3 lines create 6 angles around the point: 50°, 130°, 50°, 130°, 50°, 130°

Wait, this is still 540°. Let me reconsider.

Actually, 3 lines through a point create 6 angles that sum to 360°.

If we have angles A, B, C, A, B, C (with vertical angles equal), then 2(A + B + C) = 360°

So A + B + C = 180°. If A = 50°, then B + C = 130°.

Assuming symmetry, B = C, so 2B = 130°, so B = 65°.

Therefore, the angles are: 50°, 65°, 65°, 50°, 65°, 65°

Sum: 50° + 65° + 65° + 50° + 65° + 65° = 360° ✓

Step 6: Final verification

Total: 360° ✓

Vertical angles are equal ✓

Angles: 50°, 65°, 65°, 50°, 65°, 65°
Final answer:

The six angles measure 50°, 65°, 65°, 50°, 65°, and 65°.

Applied rules:

• Vertical Angles: Opposite angles are equal

• Angles Around Point: Sum to 360°

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

Complete Guide: Angles and Angle Relationships
\(\text{Sum of angles around point} = 360°\)
Angles Around a Point
Key definitions:

Angle: A figure formed by two rays (called sides) with a common endpoint (called vertex). Measured in degrees.

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°.

Complete methodology:
  1. Identify the Diagram: Look for intersecting lines, parallel lines, or geometric shapes
  2. Recognize Angle Pairs: Determine if angles are complementary, supplementary, vertical, or adjacent
  3. Apply Relevant Theorems: Use properties like vertical angles are equal, linear pairs are supplementary
  4. Set Up Equations: Represent relationships algebraically
  5. Solve Systematically: Use algebra to find unknown angle measures
  6. Verify Results: Check that solutions satisfy all angle relationships
Tip 1: Remember the memory aid: 'Co'mplement = 'Co'rner (90°), 'Su'pplement = 'Su'perman (180°).
Tip 2: Vertical angles look like an 'X' and are always equal.
Tip 3: Draw small arcs to mark equal angles in diagrams.
Tip 4: When solving, always verify that your answer makes geometric sense.
Common errors: Misidentifying angle relationships, confusing complementary with supplementary, forgetting that vertical angles are equal.
Exam preparation: Practice with complex diagrams containing multiple intersecting lines, work with algebraic expressions for angles, memorize key angle relationships.
Formulas to know by heart:

• Complementary Angles: ∠A + ∠B = 90°

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

• Vertical Angles: ∠A = ∠B (equal)

• Linear Pair: Adjacent angles sum to 180°

• Angles around a point: Sum = 360°

• Angles on a straight line: Sum = 180°

• Sum of angles in triangle: 180°

• Sum of angles in quadrilateral: 360°

Visualizing Angle Relationships: Types and Properties
Exercise 6: Angle Classification
Consider different angle types and their classifications:
Acute (0°-90°), Right (90°), Obtuse (90°-180°), Straight (180°)
Showing how angles relate to each other

Analysis: The chart shows how different angle types fit within the full range of 0° to 180°.

  • Acute angles are less than 90°
  • Right angles equal exactly 90°
  • Obtuse angles are between 90° and 180°
  • Straight angles equal exactly 180°

Questions & Answers

Question: How can I easily remember the difference between complementary and supplementary angles?

Answer: Here are some helpful memory aids:

For Complementary (adds to 90°):

  • 'Co'mplement = 'Co'rner (right angle of 90°)
  • 'Co'mplement = 'Co'mplete a right angle
  • The 'C' in Complementary comes before 'S' in Supplementary, just like 90 comes before 180

For Supplementary (adds to 180°):

  • 'Su'pplement = 'Su'perman (think of a straight line like Superman flying straight)
  • 'Su'pplement = 'Su'perior (180° is superior/straighter than 90°)
  • Picture a 'S'traight line of 180°

Another way: C comes before S in the alphabet, and 90° comes before 180° numerically.

Question: Are vertical angles always acute or can they be obtuse? Can you have supplementary vertical angles?

Answer: Great question! Let me clarify:

About vertical angles:

  • Vertical angles are always equal in measure
  • They can be acute (0° < angle < 90°), right (90°), or obtuse (90° < angle < 180°)
  • Vertical angles are NOT necessarily supplementary to each other

About supplementary vertical angles:

Vertical angles are only supplementary if they are both 90° each (since 90° + 90° = 180°).

When two lines intersect and form 90° angles, they are perpendicular, and all four angles are 90°.

Adjacent angles to vertical angles ARE supplementary (they form linear pairs).

Example: If one vertical angle is 40°, its vertical counterpart is also 40°, but they sum to 80°, not 180°.

Question: How do I identify adjacent angles in a complex diagram with multiple intersecting lines?

Answer: Here's a systematic approach to identify adjacent angles:

Characteristics of adjacent angles:

  • Share a common vertex (corner point)
  • Share a common side (ray)
  • Do not overlap each other
  • Have no interior points in common

Steps to identify adjacent angles:

  1. Locate the vertex where multiple lines meet
  2. Identify all the angles that share this vertex
  3. Look for pairs of angles that share a common ray
  4. Verify that the angles don't overlap

In complex diagrams, draw small arcs or color-code the angles to keep track of which ones are adjacent. Remember that each angle (except those at the ends) is adjacent to two other angles.