# The relationship between the interior and exterior angles of a polygon

**Subject** : **Mathematics**

**Mathematics**

**Class** : Basic 6 / Primary 6 /Grade 6

**Term** : Third Term / 3rd Term

**Week** : Week 4

**Topic** : The relationship between the interior and exterior angles of a polygon

### Learning Objectives:

- Students will be able to define and identify the interior and exterior angles of a polygon.
- Students will be able to use the formula to find the sum of the interior angles of a polygon.
- Students will be able to understand the relationship between an exterior angle and the two interior angles that are next to it.
- Students will be able to apply their knowledge to solve problems related to polygon angles.

**Previous Knowledge : Pupils have previous knowledge of TYPES OF ANGLES that was taught in the previous lesson**

### Materials Needed:

- Whiteboard and markers
- Printed worksheets for students
- Sample polygon shapes (triangle, square, pentagon, hexagon, octagon)
- Protractor (optional)

**Reference Materials**

- Scheme of Work
- Online Information
- Textbooks
- Workbooks
- 9 Year Basic Education Curriculum

**Content** :

- A polygon is a closed shape with three or more straight sides.
- The interior angles of a polygon are the angles formed inside the shape when you draw lines between its vertices (corners).
- The exterior angles of a polygon are the angles formed outside the shape when you extend one of its sides.
- The sum of the interior angles of a polygon with n sides is given by the formula (n-2) x 180 degrees. For example, a triangle has three sides and (3-2) x 180 = 180 degrees, a square has four sides and (4-2) x 180 = 360 degrees, and a pentagon has five sides and (5-2) x 180 = 540 degrees.
- The exterior angles of a polygon always add up to 360 degrees, regardless of how many sides it has. This means that if you add up all the exterior angles of a polygon, the sum will always be 360 degrees.
- The size of an exterior angle is equal to the sum of the two interior angles that are next to it. For example, in a triangle, each exterior angle is equal to the sum of the two interior angles that are next to it. In a square, each exterior angle is equal to 90 degrees (since each interior angle is 90 degrees), and so on

### Evaluation

- A polygon is a closed shape with ______ or more straight sides. a) two b) three c) four
- The interior angles of a polygon are the angles formed ________ the shape when you draw lines between its vertices. a) inside b) outside c) on top of
- The exterior angles of a polygon are the angles formed ________ the shape when you extend one of its sides. a) inside b) outside c) on top of
- The sum of the interior angles of a polygon with n sides is given by the formula _______ x 180 degrees. a) (n+2) b) (n-2) c) (n/2)
- The exterior angles of a polygon always add up to _______ degrees. a) 90 degrees b) 180 degrees c) 360 degrees
- The size of an exterior angle is equal to the _______ of the two interior angles that are next to it. a) difference b) sum c) product

### The relationship between the interior and exterior angles of polygon

- A triangle has three sides, and the sum of its interior angles is (3-2) x 180 = 180 degrees. The exterior angles of a triangle add up to 360 degrees, so each exterior angle is equal to (360/3) = 120 degrees. Each exterior angle of a triangle is equal to the sum of the two interior angles that are next to it, so each exterior angle is 120 degrees, which is equal to 60 + 60 degrees.
- A square has four sides, and the sum of its interior angles is (4-2) x 180 = 360 degrees. The exterior angles of a square add up to 360 degrees, so each exterior angle is equal to (360/4) = 90 degrees. Each exterior angle of a square is equal to the sum of the two interior angles that are next to it, so each exterior angle is 90 degrees, which is equal to 45 + 45 degrees.
- A pentagon has five sides, and the sum of its interior angles is (5-2) x 180 = 540 degrees. The exterior angles of a pentagon add up to 360 degrees, so each exterior angle is equal to (360/5) = 72 degrees. Each exterior angle of a pentagon is equal to the sum of the two interior angles that are next to it, so each exterior angle is 72 degrees, which is equal to 36 + 36 degrees.
- A hexagon has six sides, and the sum of its interior angles is (6-2) x 180 = 720 degrees. The exterior angles of a hexagon add up to 360 degrees, so each exterior angle is equal to (360/6) = 60 degrees. Each exterior angle of a hexagon is equal to the sum of the two interior angles that are next to it, so each exterior angle is 60 degrees, which is equal to 30 + 30 degrees.
- An octagon has eight sides, and the sum of its interior angles is (8-2) x 180 = 1080 degrees. The exterior angles of an octagon add up to 360 degrees, so each exterior angle is equal to (360/8) = 45 degrees. Each exterior angle of an octagon is equal to the sum of the two interior angles that are next to it, so each exterior angle is 45 degrees, which is equal to 22.5 + 22.5 degrees

**Evaluation**

- What are the interior angles of a polygon? a) The angles formed inside the shape when you extend one of its sides. b) The angles formed outside the shape when you draw lines between its vertices. c) The angles formed inside the shape when you draw lines between its vertices.
- What are the exterior angles of a polygon? a) The angles formed inside the shape when you extend one of its sides. b) The angles formed outside the shape when you extend one of its sides. c) The angles formed outside the shape when you draw lines between its vertices.
- What is the sum of the interior angles of a polygon with 6 sides? a) 720 degrees b) 360 degrees c) 540 degrees
- What is the sum of the exterior angles of a polygon? a) 180 degrees b) 360 degrees c) 720 degrees
- What is the relationship between an exterior angle and the two interior angles that are next to it? a) The size of an exterior angle is equal to the product of the two interior angles that are next to it. b) The size of an exterior angle is equal to the difference of the two interior angles that are next to it. c) The size of an exterior angle is equal to the sum of the two interior angles that are next to it.

Evaluation

- What are the interior angles of a polygon? a) The angles formed inside the shape when you extend one of its sides. b) The angles formed outside the shape when you draw lines between its vertices. c) The angles formed inside the shape when you draw lines between its vertices.
- What are the exterior angles of a polygon? a) The angles formed inside the shape when you extend one of its sides. b) The angles formed outside the shape when you extend one of its sides. c) The angles formed outside the shape when you draw lines between its vertices.
- What is the sum of the interior angles of a polygon with 6 sides? a) 720 degrees b) 360 degrees c) 540 degrees
- What is the sum of the exterior angles of a polygon? a) 180 degrees b) 360 degrees c) 720 degrees
- What is the relationship between an exterior angle and the two interior angles that are next to it? a) The size of an exterior angle is equal to the product of the two interior angles that are next to it. b) The size of an exterior angle is equal to the difference of the two interior angles that are next to it. c) The size of an exterior angle is equal to the sum of the two interior angles that are next to it.

### Lesson Plan Presentation

**Lesson Title**: The Relationship between the Interior and Exterior Angles of a Polygon

**Introduction (5 minutes):**

- Greet the students and introduce the topic of the lesson.
- Display a sample polygon shape on the whiteboard and ask the students if they know what it is.
- Define a polygon as a closed shape with three or more straight sides.
- Explain that in this lesson, we will learn about the interior and exterior angles of a polygon, and their relationship.

**Direct Instruction (10 minutes):**

- Define interior angles as the angles formed inside the shape when you draw lines between its vertices (corners).
- Define exterior angles as the angles formed outside the shape when you extend one of its sides.
- Show examples of the interior and exterior angles of a polygon using the sample shapes.
- Use a triangle as an example to explain that the sum of the interior angles of a polygon with n sides is (n-2) x 180 degrees.
- Demonstrate the relationship between an exterior angle and the two interior angles that are next to it by using a square as an example.

**Guided Practice (10 minutes):**

- Hand out printed worksheets for the students to complete.
- Provide a few sample polygon shapes on the worksheet and have the students label the interior and exterior angles of each shape.
- Have the students calculate the sum of the interior angles of a few sample polygon shapes using the formula (n-2) x 180 degrees.
- Provide a few questions where the students need to calculate the missing interior angle or exterior angle.
- Walk around the classroom and offer assistance to the students who need it.

**Independent Practice (15 minutes):**

- Provide a few polygon shapes on the whiteboard and have the students calculate the sum of the interior angles and the measure of each exterior angle.
- Ask the students to draw a polygon shape with a specific number of sides and label the interior and exterior angles.
- Allow the students to work independently on the worksheet or any additional questions.
- Optional: Provide protractors for the students to measure the angles.

**Closure (5 minutes):**

- Recap the main points of the lesson and ask the students to share what they learned about the interior and exterior angles of a polygon.
- Display the sample polygon shapes again and ask the students to identify the interior and exterior angles of each shape.
- Encourage the students to continue practicing and exploring polygon shapes on their own.

**Assessment**:

The assessment will be based on the student’s ability to correctly identify the interior and exterior angles of a polygon, calculate the sum of the interior angles using the formula, and demonstrate an understanding of the relationship between an exterior angle and the two interior angles that are next to it.

**Differentiation**:

- For students who are struggling, provide more guidance and additional examples.
- For students who are excelling, challenge them to draw and label complex polygons and find the sum of their interior angles