Let’s analyze the given diagram and solve for angle x.

Step 1: Understanding the Given Information

• The diagram shows multiple angles around a straight line.
• A straight line forms a total of 180°.
• The given angles are 42° and 30°.
• The two unknown angles are marked as x.

Step 2: Setting Up the Equation

Since all the angles on a straight line sum to 180°, we can write the equation as:

42° + x + x + 30° = 180°

42° + 2x + 30° = 180°

72° + 2x = 180°

2x = 180° – 72°

2x = 108°

x = 108°/2

x = 54°

Thus, the missing angle is 54°.

The Importance of Angles in Geometry

Types of Angles

1. Acute Angles: Less than 90°
2. Right Angles: Exactly 90°
3. Obtuse Angles: Greater than 90° but less than 180°
4. Straight Angles: Exactly 180°
5. Reflex Angles: Greater than 180° but less than 360°

Understanding these angles is crucial for solving geometric problems efficiently.

Angles on a Straight Line

A fundamental rule in geometry states that the sum of angles on a straight line is always 180°. This principle helps in solving many mathematical problems, including the example above.

Angles in a Triangle

The sum of the interior angles of a triangle is always 180°. This rule is useful for finding missing angles when two angles are known.

Angles in Parallel Lines

When two parallel lines are intersected by a transversal, several angle relationships arise:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Co-interior angles sum to 180°.

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