Understanding how to calculate angles in various geometric shapes is fundamental in mathematics, particularly in geometry. This section delves into the angle sum rules that are pivotal for solving problems related to angles in different shapes such as points, straight lines, triangles, and quadrilaterals. Mastery of these rules is crucial for students preparing for the Cambridge International Examination International General Certificate of Secondary Education (CIE IGCSE).

Angle Sum Rules

The angle sum rules are basic yet powerful tools in geometry. They form the foundation for solving a wide range of problems. Here’s an overview of these essential rules:

• Angles at a Point: The angles around a point sum to 360°.
• Angles on a Straight Line: The angles on a straight line sum to 180°.
• Angles in a Triangle: The angles inside a triangle sum to 180°.
• Angles in a Quadrilateral: The angles inside a quadrilateral sum to 360°.

Angles at a Point

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Rule

When multiple angles sit around a single point, their sum is 360°. This rule is particularly useful in diagrams where various lines intersect at a central point.

Example

If two angles at a point are known to be 90° and 150°, what is the measure of the third angle?

Solution:

1. Sum of angles = $360°$

2. Known angles = $90° + 150° = 240°$

3. Unknown angle = $360° - 240° = 120°$

Therefore, the measure of the third angle is 120°.

Angles on a Straight Line

Rule

The sum of angles that form a straight line is 180°. This principle is applied when you have adjacent angles that are on the same line.

Example

Given that one angle on a straight line is 110°, find the measure of the other angle.

Solution:

1. Sum of angles on a straight line = $180°$

2. Known angle = $110°$

3. Unknown angle = $180° - 110° = 70°$

Angles in a Triangle

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Rule

The interior angles of any triangle add up to 180°. This rule is fundamental for solving problems related to triangles of any type.

Example

In a triangle, if two angles are 65° and 50°, what is the measure of the third angle?

Solution:

1. Sum of angles in a triangle = $180°$

2. Known angles = $65° + 50° = 115°$

3. Unknown angle = $180° - 115° = 65°$

Angles in a Quadrilateral

Rule

The angles inside any quadrilateral sum to 360°. This is applicable to all quadrilaterals, regardless of their shape.

Example

In a quadrilateral, three angles are known to be 90°, 85°, and 95°. Find the fourth angle.

Solution:

1. Sum of angles in a quadrilateral = $360°$

2. Known angles = $90° + 85° + 95° = 270°$

3. Unknown angle = $360° - 270° = 90°$

Applying Angle Sum Rules

Problem-Solving Tips

• Identify the Shape: Determine whether you're dealing with a point, line, triangle, or quadrilateral.
• Sum the Known Angles: Add up the measures of the known angles.
• Calculate the Unknown: Use the appropriate sum rule to find the measure of the unknown angle(s).

Practice Problems

1. Problem: A triangle has angles of 45° and 75°. What is the third angle?

• Solution: The third angle $= 180° - (45° + 75°) = 60°.$

2. Problem: Find the measure of the missing angle in a quadrilateral with angles of 100°, 120°, and 30°.

• Solution: The missing angle $= 360° - (100° + 120° + 30°) = 110°.$

3. Problem: Two lines intersect at a point, creating angles of 120°, 80°, and 100°. What is the measure of the fourth angle?

• Solution: The fourth angle $= 360° - (120° + 80° + 100°) = 60°.$