Types and Relationships of Angles

Complementary Angles: Two angles whose sum is 90 degrees.
Supplementary Angles: Two angles whose sum is 180 degrees.
Adjacent Angles: Two angles that share a common side and vertex, but do not overlap.
Vertical Angles: Angles that are opposite each other when two lines intersect; they are equal.
Linear Pair: A pair of adjacent angles that form a straight line; they are supplementary.

Sum of Interior Angles:
- Formula: (n - 2) × 180°, where n = number of sides.
Sum of Exterior Angles: Always equal to 360°, regardless of the number of sides.
Regular Polygons: All interior angles and sides are equal.
- Interior Angle Formula: [ (n - 2) × 180° ] / n.

Complementary Angles:
- Example: 30° and 60° are complementary because 30° + 60° = 90°.
Supplementary Angles:
- Example: 120° and 60° are supplementary because 120° + 60° = 180°.
Applications: Used in solving problems involving right triangles and straight lines.

Definition: A line or ray that divides an angle into two equal parts.
Properties:
- The angle bisector of an angle divides it into two angles that are congruent.
Angle Bisector Theorem: In a triangle, the angle bisector divides the opposite side into segments that are proportional to the adjacent sides.

Understanding angle types aids in identifying and classifying angles in various geometrical contexts.
Angle relationships are crucial for solving problems related to parallel lines and transversals.
Polygons have specific angle-related properties that can aid in calculations and proofs.
Knowing complementary and supplementary angles is essential in various applications, including trigonometry.
Angle bisectors are foundational in triangle properties and constructions.

Acute Angle: Measures less than 90 degrees, indicating a sharp angle.
Right Angle: Measures exactly 90 degrees, commonly represented by a square in diagrams.
Obtuse Angle: Measures greater than 90 degrees but less than 180 degrees, signifying a wider angle.
Straight Angle: Measures exactly 180 degrees, appearing as a straight line.
Reflex Angle: Measures greater than 180 degrees but less than 360 degrees, indicating a larger turn.
Full Rotation: Represents a complete angle of 360 degrees, returning to the starting point.

Complementary Angles: Two angles that add up to 90 degrees; essential in forming right angles.
Supplementary Angles: Two angles that sum to 180 degrees; often relevant in straight line geometry.
Adjacent Angles: Share a common vertex and side, important for understanding angle configurations.
Vertical Angles: Formed when two lines intersect; opposite angles are always equal.
Linear Pair: Adjacent angles that form a straight line, hence supplementary.

Sum of Interior Angles: Determined using the formula (n - 2) × 180°, where n represents the number of sides, which assists in polygon classification.
Sum of Exterior Angles: Constantly equals 360° for any polygon, crucial for external angle calculations.
Regular Polygons: All sides and interior angles are equal; the interior angle of a regular polygon can be calculated with [ (n - 2) × 180° ] / n.

Complementary Angles: For instance, 30° and 60° together form a right angle by summing to 90°.
Supplementary Angles: For example, 120° and 60° form a straight line by adding up to 180°.
Applications: These concepts are vital in solving geometric problems, especially in right triangles and linear figures.

Definition: A line or ray that splits an angle into two equal parts, crucial in geometric constructions.
Properties: Ensures that the resulting angles after division are congruent.
Angle Bisector Theorem: States that in a triangle, the angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

Identifying angle types is essential for accurate classification in geometry.
Understanding angle relationships aids in solving complex problems, especially involving transversals and parallel lines.
Knowledge of polygons' angle properties supports various calculations and proofs in geometry.
Recognizing complementary and supplementary angles is critical for applications in trigonometry and geometry.
Angle bisectors play a foundational role in the study of triangles and their properties.