Page 2: Interior and Exterior Angles of Polygons

This page delves deeper into the properties of interior and exterior angles in polygons, with a focus on triangles.

Definition: Interior angle - an angle on the inside of a polygon.

Key points covered:

• The sum of interior angles in a triangle is always 180°.
• Properties of isosceles triangles: Two sides and two base angles are equal.
• Exterior angles of polygons: The sum of exterior angles in any polygon is always 360°.

Example: In a triangle with angles 58°, 80°, and x°, we can find x by solving 58° + 80° + x° = 180°.

The page includes practice problems for calculating unknown angles in various polygons.

Highlight: To find an exterior angle of a regular polygon, use the formula: 360° ÷ number of sides.

Page 3: Advanced Polygon Angle Calculations

This page focuses on more complex angle calculations in regular and irregular polygons.

Practice problems include:

1. Calculating interior angles of regular polygons with 16, 19, and 18 sides.
2. Finding missing angles in irregular polygons.
3. Determining the number of sides in polygons given the sum of interior angles.

Example: For a regular polygon with 16 sides, the size of one interior angle is $16-2$ × 180° ÷ 16 = 157.5°.

The page also covers exterior angles in polygons:

• The sum of exterior angles is always 360°.
• To find one exterior angle: 360° ÷ number of sides.

Highlight: In a regular polygon, the sum of one interior and one exterior angle is always 180°.

Page 4: Pythagorean Theorem and Its Applications

This page introduces the Pythagorean theorem and its practical applications in geometry.

Definition: Pythagorean theorem - In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides $c² = a² + b²$.

Key concepts covered:

• Using the Pythagorean theorem to find missing sides in right-angled triangles.
• Verifying if a triangle is right-angled using the theorem.
• Solving real-world problems using the Pythagorean theorem.

Example: In a right-angled triangle with sides 3cm and 4cm, the hypotenuse is √$3² + 4²$ = 5cm.

The page includes various practice problems and real-world applications, such as calculating distances in navigation.

Highlight: The Pythagorean theorem is a fundamental concept in trigonometry and has numerous practical applications in fields like construction, navigation, and physics.

This comprehensive guide provides students with a solid foundation in trigonometry and angle properties, essential for success in GCSE mathematics. By mastering these concepts and practicing with the provided examples and problems, students will be well-prepared for their exams and future mathematical studies.

Page 4: Pythagoras Theorem

This section introduces the Pythagorean theorem and its applications.

Definition: In a right-angled triangle, c² = a² + b², where c is the hypotenuse.

Example: For a triangle with sides 3cm and 4cm, the hypotenuse is √$3² + 4²$ = 5cm.

Highlight: The theorem can be used to verify if a triangle is right-angled.

Page 5: Trigonometry Introduction

This page covers basic trigonometric ratios and their applications.

Definition: SOH CAH TOA represents the three main trigonometric ratios: Sine, Cosine, and Tangent.

Vocabulary: Adjacent - the side next to the angle in question (excluding the hypotenuse).

Example: sin⁻¹, cos⁻¹, and tan⁻¹ are inverse trigonometric functions used to find angles.

Page 1: Angle Properties and Parallel Lines

This page introduces fundamental angle properties and their applications in parallel lines.

Vocabulary: Corresponding angles, alternate angles, co-interior angles, vertically opposite angles

The page covers the following key concepts:

• Angles on parallel lines: Corresponding and alternate angles are equal, while co-interior angles sum to 180°.
• Vertically opposite angles are equal.
• Properties of parallelogram angles: Opposite angles are equal, and adjacent angles sum to 180°.

Example: In a parallelogram, if one angle is 70°, the adjacent angle is 110° $180° - 70°$.

The page also introduces the concept of interior angles in polygons:

• The sum of interior angles formula: $n-2$ × 180°, where n is the number of sides.
• Examples are provided for different polygons $n = 4, 6, 20$.

Highlight: For a hexagon $n = 6$, the sum of interior angles is $6-2$ × 180° = 720°.