Lines and angles form the foundation of geometry. A line extends infinitely in both directions, a ray has one endpoint, and a line segment has two endpoints.
- Types of Angles:
- Acute angle: less than 90 degrees
- Right angle: exactly 90 degrees
- Obtuse angle: between 90 and 180 degrees
- Straight angle: exactly 180 degrees
- Reflex angle: between 180 and 360 degrees
- Pairs of Angles:
- Complementary angles: two angles whose sum is 90 degrees
- Supplementary angles: two angles whose sum is 180 degrees
- Adjacent angles: share a common vertex and side but no common interior points
- Linear pair: adjacent angles on a straight line; they are supplementary
- Vertically opposite angles: formed by two intersecting lines; they are always equal
- Parallel Lines and a Transversal:
- When a transversal cuts two parallel lines, several angle pairs are formed:
- Corresponding angles: equal (same position at each intersection)
- Alternate interior angles: equal (on opposite sides of transversal, between the parallel lines)
- Alternate exterior angles: equal (on opposite sides of transversal, outside the parallel lines)
- Co-interior (same-side interior) angles: supplementary (on same side, between the lines)
Two angles are complementary. One is 35 degrees. Find the other.
Other angle = 90 - 35 = 55 degrees.
Find the supplement of 112 degrees.
Supplement = 180 - 112 = 68 degrees.
Two lines intersect. One angle is 65 degrees. Find all four angles.
The vertically opposite angle is also 65 degrees. The adjacent angles form a linear pair, so they are 180 - 65 = 115 degrees each.
A transversal crosses two parallel lines. One co-interior angle is 70 degrees. Find the other.
Co-interior angles are supplementary: other angle = 180 - 70 = 110 degrees.
Angles in a triangle add up to 180 degrees. If two angles are 55 degrees and 75 degrees, find the third.
Third angle = 180 - 55 - 75 = 50 degrees.
Lines AB and CD are parallel. A transversal makes an angle of 130 degrees with AB. Find the alternate interior angle with CD.
Alternate interior angles are equal: 130 degrees... wait, that would be the corresponding. The alternate interior angle = 180 - 130 = 50 degrees (co-interior is supplement, alternate interior equals 130 degrees if measured from same side). Alternate interior angles are equal, so 130 degrees.
In a linear pair, one angle is three times the other. Find both angles.
Let angles be x and 3x. x + 3x = 180. 4x = 180. x = 45 degrees. Angles are 45 degrees and 135 degrees.
- Key Properties:
- Vertically opposite angles are equal.
- Corresponding angles with parallel lines are equal.
- Alternate interior angles with parallel lines are equal.
- Co-interior angles with parallel lines add to 180 degrees.
Common mistakes
Do not confuse complementary (sum = 90) and supplementary (sum = 180). When two lines intersect, remember only vertically opposite angles are equal — adjacent angles are supplementary, not equal.
Summary
Angles are classified by their measure and by how they relate to each other. Parallel lines crossed by a transversal create special equal and supplementary angle pairs. These properties are used to solve unknown angles in geometry.