Angles are all around us, from the corners of rooms to the hands of clocks. They come in different types and sizes, each with unique properties. Understanding angles helps us make sense of shapes and structures in our everyday world.

Parallel lines and transversals create special angle relationships. These connections between angles are like secret codes that unlock geometric puzzles. Knowing these patterns helps us solve problems in math and real life, from construction to navigation.

Types and Measures of Angles

Types of angles

• Angles classified based on measure in degrees
• Acute angle has measure greater than 0° and less than 90° (45°, 60°)
• Right angle has measure of exactly 90°
  • Denoted by small square at vertex (corner of a square, book)
• Obtuse angle has measure greater than 90° and less than 180° (120°, 150°)
• Straight angle has measure of exactly 180°
  • Forms straight line (horizon, table edge)

Calculation of angle measures

• Supplementary angles have measures that add up to 180°
  • If angles $a$ and $b$ are supplementary, then $a + b = 180°$ (30° and 150°, 45° and 135°)
• Complementary angles have measures that add up to 90°
  • If angles $a$ and $b$ are complementary, then $a + b = 90°$ (30° and 60°, 25° and 65°)
• Vertical angles formed by intersecting lines, directly opposite each other
  • Vertical angles have equal measures (crossing streets, an "X")
• Angle measures can be determined using a protractor

Angles Formed by Parallel Lines and a Transversal

Angles with parallel lines

• Transversal intersects two parallel lines, forming several angle relationships
• Alternate interior angles on opposite sides of transversal, inside parallel lines
  • Alternate interior angles have equal measures (Z pattern)
• Alternate exterior angles on opposite sides of transversal, outside parallel lines
  • Alternate exterior angles have equal measures (Z pattern)
• Corresponding angles on same side of transversal, one inside and one outside parallel lines
  • Corresponding angles have equal measures (F pattern)
• Same-side interior angles on same side of transversal, both inside parallel lines
  • Same-side interior angles are supplementary, add up to 180° (C pattern)

Additional Angle Concepts

Angular Measurement and Geometry

• Radian: an alternative unit for measuring angles, based on the radius of a circle
• Arc: a portion of the circumference of a circle, often used to define angle measure
• Vertex: the point where two sides of an angle meet
• Congruence: when two angles have the same measure, they are considered congruent
• Trigonometry: a branch of mathematics that studies relationships between side lengths and angles of triangles