Comprehensive Guide to Angles and Angle Pairs

Adjacent Angles

• Definition: Adjacent angles are defined as two coplanar angles that share a common vertex and a common side or ray.
• Criteria: To be considered adjacent, the interiors of the two angles must not overlap.
• Structural Components (Example):
  • Common Vertex: Point $B$
  • Common Ray: Ray $BD$
  • Resulting Adjacent Pair: $\angle ABD$ and $\angle DBC$

Complementary Angles

• Definition: These are two angles whose measures add up to a total of $90^\circ$.
• Proximity Property: Complementary angles may or may not be adjacent to one another.
• Numerical Examples of Sums:
  • $45^\circ + 45^\circ = 90^\circ$
  • $30^\circ + 60^\circ = 90^\circ$
• Recap Formula: $m \angle a + m \angle b = 90^\circ$

Supplementary Angles

• Definition: These are two angles whose measures add up to a total of $180^\circ$.
• Proximity Property: Supplementary angles may or may not be adjacent to one another.
• Numerical Examples of Sums:
  • $60^\circ + 120^\circ = 180^\circ$
  • $30^\circ + 150^\circ = 180^\circ$
• Recap Formula: $m \angle a + m \angle b = 180^\circ$

Linear Pair

• Definition: A linear pair is a pair of adjacent angles whose non-common sides form a straight line.
• Specific Relationship: All linear pairs are inherently supplementary, meaning their measures always sum to $180^\circ$.
• Numerical Examples:
  • $50^\circ$ and $130^\circ$
  • $90^\circ$ and $90^\circ$

Vertical Angles

• Definition: These are a pair of non-adjacent angles formed when two lines intersect.
• Crucial Theorem: Vertical angles are always congruent (equal in measure).
• Relationship Terms: Angles that are vertical to one another can be described as co-vertical.
• Examples based on intersecting lines:
  • $\angle 1$ and $\angle 3$ are co-vertical.
  • $\angle 2$ and $\angle 4$ are co-vertical.
• Geometric Examples (based on intersection Point $P$ and endpoints $A, B, C, D$):
  • $\angle APC$ and $\angle BPD$
  • $\angle APB$ and $\angle CPD$