Angles | Year 9 Mathematics (Cambridge) PDF Download

Angles

Types of Angles

• Acute Angle: An angle less than 90 degrees.
• Right Angle: An angle exactly 90 degrees.
• Obtuse Angle: An angle between 90 and 180 degrees.
• Straight Angle: An angle exactly 180 degrees.
• Reflex Angle: An angle more than 180 degrees but less than 360 degrees.

Angle Relationships

• Adjacent Angles: Angles that share a common side and a common vertex.
• Complementary Angles: Two angles that add up to 90 degrees.
• Supplementary Angles: Two angles that add up to 180 degrees.
• Vertically Opposite Angles: Angles opposite each other when two lines intersect. They are always equal.

Examples

• If one angle is 30 degrees, the complementary angle is 60 degrees (90 - 30).
• If one angle is 120 degrees, the supplementary angle is 60 degrees (180 - 120).

Parallel Lines and Angles

Corresponding Angles

• When two parallel lines are cut by a transversal, corresponding angles are equal.

Alternate Interior Angles

• Alternate interior angles are equal when two parallel lines are cut by a transversal.

Co-Interior Angles

• Co-interior angles are on the same side of the transversal and add up to 180 degrees.

Examples

• If two parallel lines are cut by a transversal, and one corresponding angle is 70 degrees, the other corresponding angle is also 70 degrees.
• If one alternate interior angle is 85 degrees, the other alternate interior angle is also 85 degrees.
• If one co-interior angle is 110 degrees, the other co-interior angle is 70 degrees (180 - 110).

Polygons and Angles

Interior Angles of Polygons

• The sum of the interior angles of a polygon with n sides is (n − 2) × 180 degrees.

Exterior Angles of Polygons

• The sum of the exterior angles of any polygon is always 360 degrees.

Regular Polygons

In a regular polygon (where all sides and angles are equal):

• Each interior angle is
• Each exterior angle is

Examples

• For a triangle (3 sides):
  • Sum of interior angles: (3 − 2) × 180 = 180 degrees.
  • Each interior angle in an equilateral triangle: 180/3 = 60 degrees.
• For a square (4 sides):
  • Sum of interior angles: (4 − 2) × 180 = 360 degrees.
  • Each interior angle in a regular square: 360/4 = 90 degrees.
• For a pentagon (5 sides):
  • Sum of interior angles: (5 − 2) × 180 = 540 degrees.
  • Each interior angle in a regular pentagon: 540/5 = 108 degrees.

Representing Data

Types of Data

• Categorical Data: Data that can be divided into specific categories (e.g., colors, types of animals).
• Numerical Data: Data that consists of numbers and can be discrete (countable) or continuous (measurable).

Examples

• Categorical Data: Types of fruits (apples, oranges, bananas).
• Numerical Data: Heights of students in a class.

Data Representation

• Bar Charts: Used to represent categorical data with rectangular bars.
• Histograms: Used to represent numerical data with bars that touch each other, indicating the frequency of data within certain intervals.
• Pie Charts: Circular charts divided into sectors to represent parts of a whole.
• Line Graphs: Used to represent data points over time or another continuous variable.

Examples

• Bar Chart: Showing the number of students in different classes.
• Histogram: Representing the distribution of test scores in a class.
• Pie Chart: Displaying the market share of different companies.
• Line Graph: Tracking the temperature changes over a week.