# IGCSE Higher Parallel Lines and Polygons

## Scheme of work: IGCSE Higher: Year 10: Term 2: Parallel Lines and Polygons

#### Prerequisite Knowledge

1. Understanding Angles on a Straight Line
• Concept: Knowing that the sum of angles on a straight line is $$180^\circ$$.
• Example: If two angles, $$\angle A$$ and $$\angle B$$, lie on a straight line, then: $\angle A + \angle B = 180^\circ$
2. Understanding Angles in a Triangle
• Concept: Knowing that the sum of the interior angles in a triangle is always $$180^\circ$$.
• Example: In any triangle with angles $$\angle A$$, $$\angle B$$, and $$\angle C$$: $\angle A + \angle B + \angle C = 180^\circ$

#### Success Criteria

1. Use Angle Properties of Intersecting Lines, Parallel Lines, and Angles on a Straight Line
• Objective: Students should be able to use and apply the properties of angles formed by intersecting lines, parallel lines, and angles on a straight line.
2. Calculate Interior and Exterior Angles of Regular Polygons
• Objective: Students should be able to understand the term ‘regular polygon’ and calculate the interior and exterior angles of regular polygons.
• Example: Calculate the interior angle of a regular hexagon. $\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} = \frac{(6-2) \times 180^\circ}{6} = 120^\circ$
3. Understand and Use the Angle Sum of Polygons
• Objective: Students should be able to understand and use the angle sum property of polygons.
• Example: Find the sum of the interior angles of an octagon. $\text{Sum of interior angles} = (n-2) \times 180^\circ = (8-2) \times 180^\circ = 1080^\circ$
4. Calculate the Exterior Angles of Polygons
• Objective: Students should be able to calculate the exterior angles of regular polygons and understand their properties.
• Example: Calculate the exterior angle of a regular pentagon. $\text{Exterior angle} = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ$

#### Key Concepts

1. Angle Properties and Relationships
• Concept: Understanding the fundamental properties and relationships of angles formed by intersecting lines, parallel lines, and angles on a straight line. This includes the concepts of supplementary angles (angles that add up to $$180^\circ$$), complementary angles (angles that add up to $$90^\circ$$), and corresponding, alternate, and co-interior angles formed by parallel lines and a transversal.
• Example: Knowing that vertically opposite angles are equal, such as when two lines intersect: $\text{If } \angle A = 70^\circ, \text{ then } \angle B = 70^\circ$
2. Angle Sum Properties of Polygons
• Concept: Understanding the angle sum properties of polygons, including the formulas for calculating the sum of interior angles and the measure of exterior angles in regular polygons. This includes knowing that the sum of interior angles of an $$n$$-sided polygon is $$(n-2) \times 180^\circ$$ and that the exterior angle of a regular polygon is $$\frac{360^\circ}{n}$$.
• Example: Calculating the sum of the interior angles of a hexagon: $\text{Sum of interior angles} = (6-2) \times 180^\circ = 720^\circ$

#### Common Misconceptions

1. Misinterpreting Angle Relationships
• Misconception: Students often confuse supplementary and complementary angles, or incorrectly identify corresponding, alternate, and co-interior angles when dealing with parallel lines and a transversal.
• Example: Thinking that two angles are complementary (adding up to $$90^\circ$$) when they are actually supplementary (adding up to $$180^\circ$$). $\text{Incorrectly assuming that } \angle A = 40^\circ \text{ and } \angle B = 50^\circ \text{ are supplementary, but they are complementary}.$
2. Misunderstanding Angle Sum Properties of Polygons
• Misconception: Students might incorrectly apply the formula for the sum of interior angles of a polygon, particularly confusing the number of sides $$n$$ with other values.
• Example: Miscalculating the sum of the interior angles of a hexagon by using an incorrect formula. $\text{Incorrectly calculating } \text{Sum of interior angles} = (6-1) \times 180^\circ = 900^\circ \text{ instead of } (6-2) \times 180^\circ = 720^\circ.$
3. Errors in Calculating Exterior Angles of Regular Polygons
• Misconception: Students may misunderstand that the exterior angles of a regular polygon always add up to $$360^\circ$$ and incorrectly calculate individual exterior angles.
• Example: Incorrectly calculating the exterior angle of a regular pentagon. $\text{Incorrectly calculating } \text{Exterior angle} = \frac{180^\circ}{5} = 36^\circ \text{ instead of } \frac{360^\circ}{5} = 72^\circ.$

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