Solved Exercises on Interior and Exterior Angles in Grade 7

Master interior and exterior angles: properties, calculations, and problem solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Interior Angles of Triangle
Exercise 1
In triangle ABC, angle A = 50° and angle B = 70°. Find angle C and verify that the sum of interior angles equals 180°.
Definition:

Interior Angles: The angles inside a polygon formed by two adjacent sides.

Method for triangle interior angles:
  1. Recall that interior angles of a triangle sum to 180°
  2. Add known angles
  3. Subtract from 180° to find unknown angle
  4. Verify the sum
Given
∠A = 50°, ∠B = 70°
Triangle Sum
∠A + ∠B + ∠C = 180°
Find
∠C = 60°
Step 1: Apply the triangle angle sum theorem

∠A + ∠B + ∠C = 180°

Step 2: Substitute known values

50° + 70° + ∠C = 180°

Step 3: Solve for angle C

120° + ∠C = 180°

∠C = 180° - 120° = 60°

Step 4: Verify the solution

∠A + ∠B + ∠C = 50° + 70° + 60° = 180° ✓

∠C = 60°, Sum = 180°
Final answer:

Angle C measures 60°, and the sum of interior angles is 180°.

Applied rules:

• Triangle Angle Sum: Interior angles sum to 180°

• Algebraic Solution: Isolate the unknown variable

• Verification: Check that angles sum to 180°

2 Exterior Angles of Triangle
Exercise 2
In triangle PQR, the exterior angle at vertex P measures 120°. If angle Q = 40°, find angle R and the interior angle at P.
Definition:

Exterior Angle: An angle formed by extending one side of a polygon. It is supplementary to the interior angle at that vertex.

Given
Ext∠P = 120°, ∠Q = 40°
Exterior Angle Thm
Ext∠ = sum of remote int angles
Find
∠R = 80°, ∠P = 60°
Step 1: Apply the exterior angle theorem

Exterior angle at P = ∠Q + ∠R

120° = 40° + ∠R

Step 2: Solve for angle R

∠R = 120° - 40° = 80°

Step 3: Find interior angle at P

Interior ∠P + Exterior ∠P = 180° (linear pair)

Interior ∠P = 180° - 120° = 60°

Step 4: Verify using triangle sum

∠P + ∠Q + ∠R = 60° + 40° + 80° = 180° ✓

∠R = 80°, Interior ∠P = 60°
Final answer:

Angle R measures 80° and the interior angle at P measures 60°.

Applied rules:

• Exterior Angle Theorem: Exterior angle = sum of remote interior angles

• Linear Pair: Interior + exterior = 180°

• Triangle Sum: Interior angles sum to 180°

3 Interior Angles of Quadrilateral
Exercise 3
In quadrilateral ABCD, three angles measure 80°, 100°, and 120°. Find the fourth angle and verify that the sum equals 360°.
Definition:

Quadrilateral Interior Angles: The sum of interior angles in any quadrilateral is 360°.

Given
Three angles: 80°, 100°, 120°
Quadrilateral Sum
Sum = 360°
Find
Fourth angle = 60°
Step 1: Apply the quadrilateral angle sum theorem

∠A + ∠B + ∠C + ∠D = 360°

Step 2: Substitute known values

80° + 100° + 120° + ∠D = 360°

Step 3: Calculate the sum of known angles

80° + 100° + 120° = 300°

Step 4: Solve for the fourth angle

300° + ∠D = 360°

∠D = 360° - 300° = 60°

Step 5: Verify the solution

80° + 100° + 120° + 60° = 360° ✓

Fourth angle = 60°, Sum = 360°
Final answer:

The fourth angle measures 60°, and the sum of all interior angles is 360°.

Applied rules:

• Quadrilateral Sum: Interior angles sum to 360°

• Algebraic Solution: Isolate the unknown variable

• Verification: Check that angles sum to 360°

Interior and Exterior Angle Properties
\(\text{Interior Angle Sum} = (n-2) \times 180°\)
Polygon Interior Angle Sum
Triangle
Sum = 180°
3 sides
Quadrilateral
Sum = 360°
4 sides
Pentagon
Sum = 540°
5 sides
Key definitions:

Interior Angle: The angle formed inside a polygon at each vertex by two adjacent sides.

Exterior Angle: The angle formed outside a polygon when one side is extended past a vertex.

Linear Pair: Two adjacent angles that form a straight line (sum to 180°).

Remote Interior Angles: In a triangle, the two angles not adjacent to a given exterior angle.

Regular Polygon: A polygon with all sides and all angles equal.

Complete methodology:
  1. Identify the Polygon: Determine the number of sides
  2. Select Appropriate Formula: Use correct sum formula
  3. Apply Angle Relationships: Use properties like linear pairs
  4. Solve Algebraically: Isolate unknown angles
  5. Verify Solution: Check that all angles sum correctly
  6. State Final Answer: Provide answer with proper units
Tip 1: Remember: interior + exterior angle = 180° at each vertex.
Tip 2: Exterior angles always sum to 360° regardless of the number of sides.
Tip 3: In a triangle, an exterior angle equals the sum of the two remote interior angles.
Tip 4: Draw diagrams to visualize angle relationships.
Common errors: Confusing interior and exterior angles, miscounting sides for polygon formulas, forgetting that exterior angles always sum to 360°.
Exam preparation: Memorize angle sums for triangles and quadrilaterals, practice with algebraic expressions for angles, master the exterior angle theorem.
Formulas to know by heart:

• Triangle Interior Sum: 180°

• Quadrilateral Interior Sum: 360°

• Polygon Interior Sum: (n - 2) × 180°

• Polygon Exterior Sum: 360° (always)

• Triangle Exterior Angle: Equals sum of two remote interior angles

• Linear Pair: Interior + Exterior = 180°

Solution: Exercises 4 to 5
4 Regular Pentagon Angles
Exercise 4
Find the measure of each interior and exterior angle of a regular pentagon.
Definition:

Regular Polygon: A polygon with all sides and all angles equal.

Pentagon
n = 5
Interior Angle
[(n-2)×180°]/n
Results
Int = 108°, Ext = 72°
Step 1: Calculate the sum of interior angles

Sum = (n - 2) × 180°

Sum = (5 - 2) × 180° = 3 × 180° = 540°

Step 2: Find each interior angle

In a regular polygon, all interior angles are equal

Each interior angle = 540° ÷ 5 = 108°

Step 3: Find each exterior angle

Interior + Exterior = 180° (linear pair)

Exterior angle = 180° - 108° = 72°

Step 4: Verify with exterior angle sum

Sum of exterior angles = 5 × 72° = 360° ✓

Step 5: Alternative exterior angle calculation

Each exterior angle = 360° ÷ n = 360° ÷ 5 = 72° ✓

Interior = 108°, Exterior = 72°
Final answer:

Each interior angle measures 108° and each exterior angle measures 72°.

Applied rules:

• Regular Polygon Interior: [(n - 2) × 180°] ÷ n

• Linear Pair: Interior + Exterior = 180°

• Exterior Sum: Always 360° for any polygon

5 Complex Angle Problem
Exercise 5
In a hexagon, the ratio of the six interior angles is 1:2:3:4:5:6. Find the measure of each angle.
Definition:

Angle Ratio Problems: When angles are in a given ratio, express them as multiples of a variable and use the angle sum property.

Ratio
1:2:3:4:5:6
Hexagon Sum
(6-2)×180° = 720°
Results
34.3°, 68.6°, 102.9°, 137.1°, 171.4°, 205.7°
Step 1: Set up the ratio

Let the angles be x, 2x, 3x, 4x, 5x, 6x degrees

Step 2: Apply the hexagon interior angle sum

Sum of interior angles = (n - 2) × 180°

Sum = (6 - 2) × 180° = 4 × 180° = 720°

Step 3: Set up the equation

x + 2x + 3x + 4x + 5x + 6x = 720°

21x = 720°

x = 720° ÷ 21 ≈ 34.29°

Step 4: Find each angle

Angle 1: x ≈ 34.29°

Angle 2: 2x ≈ 68.57°

Angle 3: 3x ≈ 102.86°

Angle 4: 4x ≈ 137.14°

Angle 5: 5x ≈ 171.43°

Angle 6: 6x ≈ 205.71°

Step 5: Verify the sum

34.29 + 68.57 + 102.86 + 137.14 + 171.43 + 205.71 ≈ 720° ✓

Angles: 34.3°, 68.6°, 102.9°, 137.1°, 171.4°, 205.7°
Final answer:

The six angles measure approximately 34.3°, 68.6°, 102.9°, 137.1°, 171.4°, and 205.7°.

Applied rules:

• Polygon Sum: (n - 2) × 180°

• Ratio Representation: Express as multiples of a variable

• Algebraic Solution: Solve equations with variables

Complete Guide: Interior and Exterior Angles
\(\text{Each exterior angle (regular)} = \frac{360°}{n}\)
Regular Polygon Exterior Angle
Key definitions:

Interior Angle: The angle formed inside a polygon at each vertex by two adjacent sides. The sum of interior angles depends on the number of sides.

Exterior Angle: The angle formed outside a polygon when one side is extended past a vertex. It is supplementary to the interior angle at that vertex.

Linear Pair: Two adjacent angles that form a straight line, summing to 180°. At each vertex, the interior and exterior angles form a linear pair.

Remote Interior Angles: In a triangle, the two interior angles that are not adjacent to a given exterior angle. The exterior angle equals the sum of these remote interior angles.

Regular Polygon: A polygon where all sides and all angles are equal.

Complete methodology:
  1. Identify the Polygon: Determine the number of sides (n)
  2. Select Appropriate Formula: Choose the correct formula for the property needed
  3. Apply Angle Relationships: Use properties like linear pairs or exterior angle theorem
  4. Set Up Equations: Represent relationships algebraically when needed
  5. Solve Systematically: Use algebra to find unknown values
  6. Verify Solution: Check that all angle relationships are satisfied
Tip 1: Remember that exterior angles of any polygon always sum to 360°, regardless of the number of sides.
Tip 2: For regular polygons, divide the total sum by the number of sides to find each angle.
Tip 3: The interior and exterior angles at each vertex are supplementary (sum to 180°).
Tip 4: Draw and label diagrams to visualize angle relationships and verify your solutions.
Common errors: Confusing interior and exterior angles, miscounting sides when applying formulas, forgetting that exterior angles always sum to 360°, arithmetic mistakes in algebraic equations.
Exam preparation: Memorize the basic angle sums (triangle = 180°, quadrilateral = 360°), practice with both regular and irregular polygons, master the exterior angle theorem for triangles.
Formulas to know by heart:

• Triangle Interior Sum: 180°

• Quadrilateral Interior Sum: 360°

• Polygon Interior Sum: (n - 2) × 180°

• Polygon Exterior Sum: 360° (always)

• Regular Polygon Interior: [(n - 2) × 180°] ÷ n

• Regular Polygon Exterior: 360° ÷ n

• Triangle Exterior Angle: Equals sum of two remote interior angles

• Linear Pair: Interior + Exterior = 180°

Visualizing Angle Relationships: Interior vs Exterior
Exercise 6: Angle Sum Patterns
Consider how interior and exterior angle sums change with the number of sides:
Triangle (n=3) to Octagon (n=8)
Showing the patterns for both types of angles

Analysis: The chart shows how interior and exterior angle sums behave with different numbers of sides.

  • Interior angle sum increases with more sides
  • Exterior angle sum remains constant at 360°
  • Individual interior angles increase in regular polygons
  • Individual exterior angles decrease in regular polygons

Questions & Answers

Question: Why do exterior angles always sum to 360°, no matter how many sides the polygon has?

Answer: This is a beautiful geometric property! Think of it this way:

Conceptual explanation:

  • Imagine walking around the perimeter of any polygon
  • At each vertex, you turn through the exterior angle
  • After completing one full trip around the polygon, you've turned a complete rotation
  • A complete rotation is always 360°, regardless of the path shape

Mathematical proof:

  • At each vertex: Interior angle + Exterior angle = 180°
  • Sum of all interior + exterior angles = n × 180°
  • Sum of interior angles = (n - 2) × 180°
  • Therefore: Sum of exterior angles = n × 180° - (n - 2) × 180° = 360°

Question: How do I know when to use the interior angle sum formula vs. the exterior angle sum formula?

Answer: Here's how to decide which formula to use:

Use Interior Angle Sum when:

  • You need to find the sum of all interior angles
  • You know some interior angles and need to find the remaining ones
  • Working with problems involving the inside of polygons
  • Dealing with regular polygons to find each interior angle

Use Exterior Angle Sum when:

  • You need to find the sum of all exterior angles (always 360°)
  • Working with exterior angles specifically
  • Using the exterior angle theorem for triangles
  • Finding individual exterior angles in regular polygons

Remember: Interior + Exterior = 180° at each vertex (linear pair).

Question: What is the exterior angle theorem for triangles and how do I use it?

Answer: The exterior angle theorem is a powerful tool for triangles:

Exterior Angle Theorem:

In any triangle, the measure of an exterior angle equals the sum of the measures of the two remote interior angles.

How to use it:

  • Identify the exterior angle you're interested in
  • Identify the two interior angles that are NOT adjacent to that exterior angle (remote angles)
  • Apply the formula: Exterior angle = Remote interior angle 1 + Remote interior angle 2

Example: In triangle ABC, if you extend side BC to form an exterior angle at C, then that exterior angle equals ∠A + ∠B.

This theorem is especially useful when you need to find an unknown angle without calculating other angles first.