Lines and Angles
6.1 Lines and Angles
6.2 Triangles
6.3 Polygons
6.4 Circles
6.5 Key Formulas
Lines and Angles
A line is a collection of points that extends without limit in a straight formation. A line can be named by a single letter, like line l, or it can be named according to two points that it contains, like line AB. The second way of naming a line indicates an important property common to all lines: any two points in space determine a line. For example, given two points, J and K:
a line is determined:
This line is called JK.
Line Segments
A line segment is a section of a line. It is named and determined by its endpoints. Unlike a line, whose length is infinite, a line segment has finite length. Line segment AB is pictured below.
Midpoint of a Line Segment
The midpoint of a line segment is the point on the segment that is equidistant (the same distance) from each endpoint. Because a midpoint splits a line segment into two equal halves, the midpoint is said to bisect the line segment.
Technically speaking, an angle is the union of two rays (lines that extend infinitely in just one direction) that share an endpoint (called the vertex of the angle). The measure of an angle is how far you must rotate one of the rays such that it coincides with the other.
In this guide, and for the Math IIC, you don’t need to bother with such a technical definition. Suffice it to say, angles are used to measure rotation. One full revolution around a point creates an angle of 360 degrees, or 360º. A half revolution, also known as a straight angle, is 180º. A quarter revolution, or right angle, is 90º.
In text, angles can also be indicated through the symbol .
Vertical Angles
When two lines or line segments intersect, two pairs of congruent (equal) angles are created. The angles in each pair of congruent angles created by the intersection of two lines are called vertical angles:
In this figure, angles 1 and 4 are vertical angles (and therefore congruent), as are angles 2 and 3.
Supplementary and Complementary Angles
Supplementary angles are two angles that together add up to 180º. Complementary angles are two angles that add up to 90º.
Whenever you have vertical angles, you also have supplementary angles. In the diagram of vertical angles above, 1 and 2, 1 and 3, 2 and 4, and 3and 4 are all pairs of supplementary angles.
Parallel Lines
Lines that don’t intersect are called parallel lines. The intersection of one line with two parallel lines creates many interesting angle relationships. This situation is often referred to as “parallel lines cut by a transversal,” where the transversal is the nonparallel line. As you can see in the diagram below of parallel lines AB and CD and transversal EF, two parallel lines cut by a transversal will form 8 angles.
Among the eight angles formed, three special angle relationships exist:
  1. Alternate exterior angles are pairs of congruent angles on opposite sides of the transversal, outside of the space between the parallel lines. In the figure above, there are two pairs of alternate exterior angles: 1 and 8, and 2and 7.
  2. Alternate interior angles are pairs of congruent angles on opposite sides of the transversal in the region between the parallel lines. In the figure above, there are two pairs of alternate interior angles: 3 and 6, and 4 and 5.
  3. Corresponding angles are congruent angles on the same side of the transversal. Of two corresponding angles, one will always be between the parallel lines, while the other will be outside the parallel lines. In the figure above, there are four pairs of corresponding angles: 1 and 5, 2 and 6, 3 and 7, and 4 and 8.
In addition to these special relationships between angles formed by two parallel lines cut by a transversal, all adjacent angles are supplementary.
Perpendicular Lines
Two lines that intersect to form a right (90º) angle are called perpendicular lines. Line segments AB and CD are perpendicular.
A line or line segment is called a perpendicular bisector when it intersects a line segment at the midpoint, forming vertical angles of 90º in the process. For example, in the above figure, since AD = DB, CD is the perpendicular bisector of AB.
Keep in mind that if a single line or line segment is perpendicular to two different lines or line segments, then those two lines or line segments are parallel. This is actually just another example of parallel lines being cut by a transversal (in this case, the transversal is perpendicular to the parallel lines), but it is a common situation when dealing with polygons.
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