Where Lines Meet: Angles & Lines
Angles are formed where two rays share a common endpoint, and lines create predictable angle relationships when they intersect or run parallel. Understanding angle types, angle pairs, and the properties of parallel and perpendicular lines is fundamental to geometry, from constructing buildings to designing circuits and navigating with bearings.
Components of Angles & Lines
This section covers the key angle and line relationships:
- Angle Types: Acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), and straight (exactly 180°).
- Complementary & Supplementary Angles: Complementary angles add to 90°; supplementary angles add to 180°.
- Vertical & Adjacent Angles: Vertical angles (formed by intersecting lines) are always equal; adjacent angles share a common side.
- Parallel Lines & Transversals: When a transversal crosses parallel lines, it creates corresponding, alternate interior, and alternate exterior angle pairs that are equal.
Examples of Angles & Lines
Angle Types Examples
- An angle measuring 35° is acute because it is less than 90°.
- A corner of a book forms a 90° right angle.
- An angle measuring 140° is obtuse because it is between 90° and 180°.
Complementary & Supplementary Examples
- Two angles measure 55° and 35°. Since 55 + 35 = 90, they are complementary.
- Two angles measure 110° and 70°. Since 110 + 70 = 180, they are supplementary.
- If one angle is 42°, its complement is 90 - 42 = 48° and its supplement is 180 - 42 = 138°.
Vertical & Adjacent Examples
- Two intersecting lines form four angles. If one angle is 65°, the vertical angle across from it is also 65°, and each adjacent angle is 180 - 65 = 115°.
- At a road intersection, opposite angles are vertical angles and always equal.
- Two adjacent angles on a straight line always add to 180° (they form a linear pair).
Parallel Lines & Transversals Examples
- A transversal crosses two parallel lines creating a 50° angle at the first line. The corresponding angle at the second line is also 50°.
- Alternate interior angles formed by a transversal and parallel lines are equal: if one is 72°, the other is 72°.
- A transversal creates a 115° angle with one parallel line. The co-interior (same-side interior) angle at the other line is 180 - 115 = 65°.