Mathematics Enhancement and Consolidation Camps Lesson 27

Solving Problems involving Sides and Angles of a Polygon

1. Solving Problems involving Sides and Angles of a Polygon Mathematics 7 Lesson 27 Enhancement and Consolidation Camps

2. Lesson Component 1 (Lesson Short Review) Time: 7 minutes Questions Key Idea: Solving Problems involving Sides and Angles of a Polygon . ( i ) Complete: A polygon with equal sides and equal interior angles is called a ………. polygon. (ii) Complete: An interior angle and associated exterior angle of a polygon add to ………. degrees regular 180

3. Lesson Component 1 (Lesson Short Review) Time: 7 minutes Questions Key Idea: Solving Problems involving Sides and Angles of a Polygon . 2. Use the formula: Measure of exterior angle of regular polygon = 360° 𝑛𝑛 , where 𝑛𝑛 is the number of sides of the polygon, to find the size of the exterior angles of: an equilateral triangle. (ii) a regular nonagon (9-sided polygon) 120° 40°

4. Lesson Component 1 (Lesson Short Review) Time: 7 minutes Questions Key Idea: Solving Problems involving Sides and Angles of a Polygon . 3. The exterior angles of a regular dodecagon (12-sided polygon) are each 30°. Use this information to calculate the sum of the interior angles of a dodecagon 1800°

5. corresponding, dodecagon/nonagon/octagon, exterior angle, formula, interior angle, property, regular polygon Lesson Component 3 (Lesson Language Practice) Time: 5 minutes Direction: Choose the correct word that corresponds to the following statements. _____________________angles are two angles in the same relative position on two lines when those lines are cut by a transversal. _____________________ a pair of parallel lines that are intersected with a transversal. _____________________is an equation or rule that shows a relationship between two or more numbers. _____________________angle at a vertex of a polygon, the angle that lies between the polygon. _____________________is a polygon in which all the angles are equal and all of the sides are equal. _____________________is a twelve sided polygon. _____________________is a nine sided polygon. _____________________is a polygon with eight sides. corresponding exterior angle formula interior angle regular polygon dodecagon nonagon octagon

6. Lesson Component 4 (Lesson Activity) Time: 25 minutes Part 4A

7. Key Idea: Solving Problems involving Sides and Angles of a Polygon . Questions: Matthew observes that Triangle 1 and Triangle 6 have two sides each that are sides of the octagon. By considering their sides, what type of triangle must Triangle 1 and Triangle 6 be? (ii) How many pairs of equal corresponding sides and angles do Triangle 1 and Triangle 6 have? isosceles three pairs of corresponding sides and three pairs of corresponding angles Part 4B

8. 2. Matthew knows the formula: Measure of exterior angle of regular polygon = 360° 𝑛𝑛 , where 𝑛𝑛 is the number of sides of the polygon. ( i ) The angle marked 𝑎𝑎° in the diagram is one of the exterior angles of the octagon. Use the formula to show that 𝑎𝑎° = 45°. (ii) Each of the six equal angles that meet at 𝐴𝐴 is an angle of each of Triangles 1 to 6. By first determining the size of each interior angle of the octagon using the result in 2( i ), find the size of each of the six angles. Exterior angle of octagon =360° 8 = 45° Interior angle of octagon = 180° − 45° = 135° Size of six equal angles that meet at 𝐴𝐴 = 135° 6 = 22.5°

9. 3. ( i ) Using the size of each interior angle found in 2(ii), show that the sum of the interior angles of the octagon is 1080°. (ii) How could Matthew have found the sum of the interior angles of the octagon, without knowing the size of each interior angle, immediately following the construction of his diagram? Sum of the interior angles = 8 × 135° = 1080° Sum of the interior angles = angle sum of the six triangles that form the octagon = 6 × 180° = 1080°

10. Key Idea: Solving Problems involving Sides and Angles of a Polygon . Questions: 1. Triangle 3 is a right-angled triangle. If the diagonal 𝐴D of the octagon is approximately 4.8 meters long, find the area of Triangle 3. 2. Using the fact that Triangle 3 is right-angled and results found in 2(ii) of Part 4B, calculate the size of the obtuse angle in Triangle 2. Area of Triangle 3 ≅ 28.8   Size of obtuse angle in Triangle 2 = 112.5° Part 4C

11. Key Idea: Solving Problems involving Sides and Angles of a Polygon . 3. The formula that is used to find the number of diagonals in any polygon is: Number of diagonals = 1 2 × 𝑛𝑛 × (𝑛𝑛 − 3), where '𝑛𝑛' represents the number of sides of the polygon. Use the formula to find the number of diagonals that Matthew could draw in the octagon. (ii) What fraction of the number of possible diagonals did Matthew draw when he divided the octagon into 6 triangles? 20   Part 4C

12. . Lesson Component 5 (Lesson Conclusion – Reflection/Metacognition on Student Goals) Time: 5 minutes o What do you think were the key mathematical concepts addressed in this lesson? o Would you rate your level of understanding of the material covered in this lesson as high, moderate, or low? o Has the lesson helped you gain further insight into aspects of the material covered that represent strengths or represent weaknesses?

13. . Lesson Component 5 (Lesson Conclusion – Reflection/Metacognition on Student Goals) Time: 5 minutes o What would you describe as the main barriers, if any, to your ongoing progress and achievement in relation to the topic area addressed in this lesson? o What do you think would best assist your ongoing progress and achievement in relation to the topic area?