11-Plus Mathematics: Angles

Understanding Angles

In 11-plus Mathematics, a crucial topic is angles. This section covers the measurement, calculation, and properties of angles, which are fundamental in geometry.

Types of Angles

Angles can be classified into several types:

• Acute Angle: An angle less than 90°.
• Right Angle: An angle exactly equal to 90°.
• Obtuse Angle: An angle greater than 90° but less than 180°.
• Straight Angle: An angle exactly equal to 180°.
• Reflex Angle: An angle greater than 180° but less than 360°.

Measuring Angles

To measure angles accurately, a protractor is used. Here’s how to use a protractor:

1. Place the midpoint of the protractor at the vertex of the angle.
2. Align one side of the angle with the zero line of the protractor.
3. Read the measurement on the scale where the other side of the angle intersects.

Calculating Missing Angles

Understanding how to calculate missing angles is essential:

Angles on a Straight Line

The sum of angles on a straight line is always 180°. For example, if one angle measures 120°, the missing angle can be calculated as:

Worked Example

Problem: Find the missing angle on a straight line if one angle is 120°.

Solution:

• Let the missing angle be x.
• According to the rule: 120° + x = 180°
• Rearranging gives: x = 180° - 120° = 60°

Angles Around a Point

The sum of angles around a point is always 360°. For example:

Worked Example

Problem: If three angles around a point measure 90°, 120°, and 30°, find the missing angle.

Solution:

• Let the missing angle be y.
• According to the rule: 90° + 120° + 30° + y = 360°
• Rearranging gives: y = 360° - (90° + 120° + 30°) = 120°

Angles in Shapes

Understanding angles in triangles and quadrilaterals is also essential:

• Triangles: The sum of angles in a triangle is always 180°.
• Quadrilaterals: The sum of angles in a quadrilateral is always 360°.

Vertically Opposite Angles

When two lines intersect, the angles opposite each other are called vertically opposite angles and are always equal.

Angles in Parallel Lines

When a transversal crosses parallel lines, several angle relationships occur, including:

• Corresponding Angles: Equal in measure.
• Alternate Angles: Equal in measure.
• Co-interior Angles: Sum to 180°.

Understanding these concepts will greatly enhance your ability to tackle angle-related questions in the 11-plus exam.