2.1 Points, Lines, Planes & Basic Notation

Undefined Terms: Points, Lines, and Planes

Every theorem you will meet on the Geometry Regents traces back to three undefined terms — point, line, and plane. New York's Next Generation standard G-CO.A.1 expects you to know the precise definitions built from these terms, so start here.

A point has no size; it only marks a location and is named with a capital letter (point A). A line extends forever in two directions, has no thickness, and is named by any two points on it with a double-arrow symbol, such as line AB, or by a single lowercase letter. A plane is a flat surface that extends forever in every direction; name it with three points that are not on the same line, or a single capital letter.

Two useful vocabulary words: points are collinear when they lie on one line, and points are coplanar when they lie in one plane. Two key postulates: through any two points there is exactly one line, and through any three non-collinear points there is exactly one plane.

Segments, Rays, and Angles

A line segment is the part of a line between two endpoints, including them; segment AB has a definite length, written AB with no bar. A ray starts at one endpoint and extends forever through a second point; ray AB begins at A, so the order of letters matters. When two rays share an endpoint, they form an angle; the shared endpoint is the vertex and the rays are the sides.

Notice the vertex letter always sits in the middle of an angle name, so ∠ABC and ∠CBA are the same angle with vertex B.

Opposite Rays and Betweenness

Two rays that share an endpoint and point in exactly opposite directions are opposite rays; together they form a straight line and a straight (180°) angle. When one point lies on the segment connecting two others, we say it is between them - a precise idea that the Segment Addition Postulate depends on. Distance along a line is always a non-negative number, so a length such as AB can never be negative even when the coordinates are.

A subtle-but-tested distinction: equal and congruent are not written the same way. AB = CD compares two numbers (the lengths are equal), while segment AB ≅ segment CD compares two figures. The same holds for angles: m∠A = m∠B compares measures, while ∠A ≅ ∠B compares the angles themselves. On the Regents you use the congruence symbol for figures and the equals sign for their measures.

Classifying Angles by Measure

Angle measures on the Regents run from 0° up to 360°. Memorize the four categories you will label on diagrams:

Segment and Angle Addition Postulates

Two "whole = sum of parts" postulates drive most numeric problems in this chapter.

• Segment Addition Postulate: if B is between A and C, then AB + BC = AC.
• Angle Addition Postulate: if ray BD lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Worked example. B is between A and C. If AB = 2x + 1, BC = 3x - 4... actually BC = 3x minus 4, and AC = 27, find AB. Write 2x + 1 + (3x minus 4) = 27, so 5x minus 3 = 27, giving 5x = 30 and x = 6. Then AB = 2(6) + 1 = 13 and BC = 3(6) minus 4 = 14; the check 13 + 14 = 27 confirms it.

Midpoints and Bisectors

A midpoint M of segment AB splits it into two congruent halves, so AM ≅ MB and each equals one-half of AB. A segment bisector is any line, ray, or segment that passes through the midpoint. An angle bisector is a ray from the vertex that divides an angle into two congruent angles.

Worked example. Ray BD bisects ∠ABC. If m∠ABC = 6x and m∠ABD = 2x + 20, find m∠ABC. Because BD bisects the angle, m∠ABD equals half of m∠ABC, which is 3x. Set 2x + 20 = 3x to get x = 20, so m∠ABC = 6(20) = 120°.

Reading Diagram Marks

Regents diagrams communicate with tick marks and arcs, so learn to read them. Matching tick marks on segments mean the segments are congruent; matching arcs inside angles mean the angles are congruent; a small square at a vertex means a right angle; and arrowheads on two lines mean the lines are parallel. Never assume a fact from how a figure "looks" — trust only the marks, the given information, and stated measures. A diagram may be drawn out of scale on purpose, so a "wide" angle is not automatically obtuse and two segments that appear equal are not congruent unless they carry matching tick marks.