# IGCSE Mathematics Foundation: Angle Geometry

## Scheme of work: IGCSE Foundation: Year 10: Term 3: Angle Geometry

#### Prerequisite Knowledge

1. Knowing the Names of 2D Shapes
• Concept: Familiarity with the names and basic properties of common 2D shapes such as squares, rectangles, triangles, circles, and polygons.
• Example: $\text{Square, Rectangle, Triangle, Circle, Hexagon, etc.}$
2. Understanding the Terms Parallel and Perpendicular
• Concept: Knowing that parallel lines are lines in a plane that never meet, no matter how far they are extended. Perpendicular lines are lines that intersect at a right angle (90 degrees).
• Example: $\text{Parallel lines: } AB \parallel CD$ $\text{Perpendicular lines: } AB \perp CD$
3. Identifying Different Types of Angles
• Concept: Recognizing and naming different types of angles including acute, obtuse, right, and straight angles.
• Example: $\text{Acute angle: less than 90 degrees}$ $\text{Right angle: exactly 90 degrees}$ $\text{Obtuse angle: between 90 and 180 degrees}$ $\text{Straight angle: exactly 180 degrees}$

#### Success Criteria

1. Understand and Use the Relationships Between Angles on a Straight Line
• Objective: Recognize that angles on a straight line add up to 180 degrees and use this relationship to find unknown angles.
• Example: $\text{If } \angle A + \angle B = 180^\circ, \text{ and } \angle A = 120^\circ, \text{ then } \angle B = 60^\circ.$
2. Understand and Use the Relationships Between Angles at a Point
• Objective: Recognize that angles around a point add up to 360 degrees and use this relationship to find unknown angles.
• Example: $\text{If } \angle A + \angle B + \angle C = 360^\circ, \text{ and } \angle A = 120^\circ \text{ and } \angle B = 150^\circ, \text{ then } \angle C = 90^\circ.$
3. Understand and Use Vertically Opposite Angles
• Objective: Recognize that vertically opposite angles are equal and use this relationship to find unknown angles.
• Example: $\text{If } \angle A = 45^\circ, \text{ then the vertically opposite angle } \angle B = 45^\circ.$
4. Understand and Use the Relationships Between Angles Formed Within Parallel Lines
• Objective: Recognize and use corresponding angles, alternate angles, and co-interior angles to find unknown angles in parallel lines.
• Example: $\text{If } \angle A = 70^\circ \text{ (corresponding angle), then } \angle B = 70^\circ.$ $\text{If } \angle C = 110^\circ \text{ (alternate angle), then } \angle D = 110^\circ.$ $\text{If } \angle E = 80^\circ \text{ (co-interior angle), then } \angle F = 100^\circ \text{ (because } \angle E + \angle F = 180^\circ \text{)}.$
5. Recognize and Use the Angle Properties of Triangles
• Objective: Recognize that the sum of the angles in a triangle is 180 degrees and use this relationship to find unknown angles.
• Example: $\text{If } \angle A + \angle B + \angle C = 180^\circ, \text{ and } \angle A = 50^\circ \text{ and } \angle B = 60^\circ, \text{ then } \angle C = 70^\circ.$
6. Recognize and Use the Properties of Squares, Rectangles, Parallelograms, Trapezia, and Rhombuses
• Objective: Recognize and use the angle properties and other characteristics of squares, rectangles, parallelograms, trapezia, and rhombuses to solve problems.
• Example: $\text{A square has four equal angles of } 90^\circ.$ $\text{A rectangle has opposite angles equal and all angles are } 90^\circ.$ $\text{A parallelogram has opposite angles equal and adjacent angles sum up to } 180^\circ.$ $\text{A trapezium has one pair of parallel sides and the sum of the interior angles is } 360^\circ.$ $\text{A rhombus has all sides equal and opposite angles are equal.}$

#### Key Concepts

• Understand and Use the Relationships Between Angles on a Straight Line
• Concept: Angles on a straight line add up to 180 degrees. This concept helps in finding unknown angles when given the measure of one or more angles on the line.
• Example: $\text{If } \angle A + \angle B = 180^\circ, \text{ and } \angle A = 120^\circ, \text{ then } \angle B = 60^\circ.$
• Understand and Use the Relationships Between Angles at a Point
• Concept: Angles around a point add up to 360 degrees. This concept is useful for determining unknown angles when given several angles that meet at a common point.
• Example: $\text{If } \angle A + \angle B + \angle C = 360^\circ, \text{ and } \angle A = 120^\circ \text{ and } \angle B = 150^\circ, \text{ then } \angle C = 90^\circ.$
• Understand and Use Vertically Opposite Angles
• Concept: Vertically opposite angles are equal. This concept helps in finding unknown angles formed by the intersection of two lines.
• Example: $\text{If } \angle A = 45^\circ, \text{ then the vertically opposite angle } \angle B = 45^\circ.$
• Understand and Use the Relationships Between Angles Formed Within Parallel Lines
• Concept: Corresponding angles, alternate angles, and co-interior angles have specific relationships when lines are parallel. This concept is critical for solving problems involving parallel lines.
• Example: $\text{If } \angle A = 70^\circ \text{ (corresponding angle), then } \angle B = 70^\circ.$ $\text{If } \angle C = 110^\circ \text{ (alternate angle), then } \angle D = 110^\circ.$ $\text{If } \angle E = 80^\circ \text{ (co-interior angle), then } \angle F = 100^\circ \text{ (because } \angle E + \angle F = 180^\circ \text{)}.$
• Recognize and Use the Angle Properties of Triangles
• Concept: The sum of the angles in a triangle is always 180 degrees. This concept is fundamental for solving problems involving unknown angles in triangles.
• Example: $\text{If } \angle A + \angle B + \angle C = 180^\circ, \text{ and } \angle A = 50^\circ \text{ and } \angle B = 60^\circ, \text{ then } \angle C = 70^\circ.$
• Recognize and Use the Properties of Squares, Rectangles, Parallelograms, Trapezia, and Rhombuses
• Concept: Each type of quadrilateral has specific angle properties and other characteristics that are useful for solving geometric problems.
• Example: $\text{A square has four equal angles of } 90^\circ.$ $\text{A rectangle has opposite angles equal and all angles are } 90^\circ.$ $\text{A parallelogram has opposite angles equal and adjacent angles sum up to } 180^\circ.$ $\text{A trapezium has one pair of parallel sides and the sum of the interior angles is } 360^\circ.$ $\text{A rhombus has all sides equal and opposite angles are equal.}$
• #### Common Misconceptions

1. Understand and Use the Relationships Between Angles on a Straight Line
• Common Mistake: Forgetting that angles on a straight line must add up to 180 degrees.
• Example: $\text{Incorrect: } \angle A = 120^\circ \text{ and } \angle B = 70^\circ \text{ (sum is 190 degrees, which is incorrect).}$ $\text{Correct: If } \angle A = 120^\circ, \text{ then } \angle B = 60^\circ \text{ (sum is 180 degrees).}$
2. Understand and Use the Relationships Between Angles at a Point
• Common Mistake: Forgetting that angles around a point must add up to 360 degrees.
• Example: $\text{Incorrect: } \angle A = 120^\circ, \angle B = 150^\circ, \text{ and } \angle C = 100^\circ \text{ (sum is 370 degrees, which is incorrect).}$ $\text{Correct: If } \angle A = 120^\circ \text{ and } \angle B = 150^\circ, \text{ then } \angle C = 90^\circ \text{ (sum is 360 degrees).}$
3. Understand and Use Vertically Opposite Angles
• Common Mistake: Confusing vertically opposite angles with adjacent angles.
• Example: $\text{Incorrect: If } \angle A = 45^\circ, \text{ then assuming the adjacent angle } \angle B = 45^\circ \text{ (instead of the opposite angle).}$ $\text{Correct: If } \angle A = 45^\circ, \text{ then the vertically opposite angle } \angle B = 45^\circ.$
4. Understand and Use the Relationships Between Angles Formed Within Parallel Lines
• Common Mistake: Misidentifying corresponding, alternate, and co-interior angles.
• Example: $\text{Incorrect: If } \angle A = 70^\circ \text{ (corresponding angle), then mistakenly stating } \angle B = 110^\circ.$ $\text{Correct: If } \angle A = 70^\circ \text{ (corresponding angle), then } \angle B = 70^\circ.$ $\text{Incorrect: If } \angle C = 110^\circ \text{ (alternate angle), then mistakenly stating } \angle D = 70^\circ.$ $\text{Correct: If } \angle C = 110^\circ \text{ (alternate angle), then } \angle D = 110^\circ.$ $\text{Incorrect: If } \angle E = 80^\circ \text{ (co-interior angle), then mistakenly stating } \angle F = 80^\circ.$ $\text{Correct: If } \angle E = 80^\circ \text{ (co-interior angle), then } \angle F = 100^\circ \text{ (because } \angle E + \angle F = 180^\circ \text{)}.$
5. Recognize and Use the Angle Properties of Triangles
• Common Mistake: Forgetting that the sum of the angles in a triangle is always 180 degrees.
• Example: $\text{Incorrect: If } \angle A = 50^\circ \text{ and } \angle B = 60^\circ, \text{ then assuming } \angle C = 80^\circ \text{ (sum is 190 degrees, which is incorrect).}$ $\text{Correct: If } \angle A = 50^\circ \text{ and } \angle B = 60^\circ, \text{ then } \angle C = 70^\circ \text{ (sum is 180 degrees).}$
6. Recognize and Use the Properties of Squares, Rectangles, Parallelograms, Trapezia, and Rhombuses
• Common Mistake: Confusing the properties of different quadrilaterals.
• Example: $\text{Incorrect: Assuming all angles in a parallelogram are } 90^\circ \text{ (instead of knowing that opposite angles are equal and adjacent angles sum up to } 180^\circ\text{).}$ $\text{Correct: A rectangle has opposite angles equal and all angles are } 90^\circ.$ $\text{Correct: A parallelogram has opposite angles equal and adjacent angles sum up to } 180^\circ.$ $\text{Correct: A trapezium has one pair of parallel sides and the sum of the interior angles is } 360^\circ.$ $\text{Correct: A rhombus has all sides equal and opposite angles are equal.}$

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