1.2 Points, lines, planes, and angles

Geometry's building blocks are points, lines, and planes. These undefined terms can't be formally defined using simpler geometric concepts, but they have precise descriptions that everything else in geometry builds on. Angles, formed where two rays share an endpoint, give you the tools to measure and describe the relationships between those building blocks.

Fundamentals of Geometry

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Points, lines, and planes

A point represents an exact location in space. It has no size, no width, no dimension at all. You label points with a single uppercase letter (A, P).

• Two distinct points determine exactly one line.
• Any single point can lie on infinitely many different lines and planes.

A line is a straight path that extends infinitely in both directions. It has length but no thickness. You can name a line with a lowercase letter ($\ell$) or by two points on it ($\overleftrightarrow{AB}$).

• A line contains infinitely many points.
• Two distinct points uniquely define a line. This means if you pick any two points, there's exactly one line passing through both.
• Parallel lines lie in the same plane and never intersect.
• Skew lines do not intersect and are not parallel. They exist in different planes. You'll only encounter skew lines in three-dimensional figures, never in a single flat plane.

A plane is a flat, two-dimensional surface that extends infinitely in all directions. Like a line, it has no thickness. You label a plane with a single uppercase letter (plane $P$) or by three non-collinear points (plane $ABC$).

• A plane contains infinitely many points and infinitely many lines.
• Three ways to uniquely define a plane:
  1. Three non-collinear points (if the points were collinear, infinitely many planes could pass through them)
  2. A line and a point not on that line
  3. Two distinct parallel lines

Classification of angles

An angle forms when two rays share a common endpoint called the vertex. You measure angles in degrees, where a full rotation equals 360°. Standard notation is $\angle ABC$ or $\angle B$, where $B$ is the vertex. When three letters are used, the vertex is always the middle letter.

Types by measure:

• Acute angle: greater than 0° and less than 90° (e.g., 45°, 60°)
• Right angle: exactly 90°, formed by perpendicular lines. Marked with a small square at the vertex.
• Obtuse angle: greater than 90° but less than 180° (e.g., 120°, 150°)
• Straight angle: exactly 180°, forming a straight line
• Reflex angle: greater than 180° but less than 360° (e.g., 270°, 330°)

Angle pair relationships:

• Complementary angles sum to 90°. If one angle measures 30°, its complement measures 60°. The two angles do not need to be adjacent.
• Supplementary angles sum to 180°. If one angle measures 60°, its supplement measures 120°. Again, they don't need to be adjacent.
• Vertical angles are the non-adjacent pairs formed when two lines intersect. They are always congruent. So if two lines cross and one angle measures 70°, the angle directly across from it also measures 70°.
• Adjacent angles share a common vertex and a common side but do not overlap (no common interior points). When adjacent angles together form a straight line, they are supplementary.

Collinearity and coplanarity concepts

Collinear points all lie on the same line. For example, if points $A$, $B$, and $C$ all sit on line $\ell$, they are collinear. Two points are always collinear (since they define a line), so collinearity is really a meaningful test only for three or more points.

Coplanar points, lines, or other objects all lie in the same plane. For example, line $\ell$ and point $P$ are coplanar if $P$ lies on a plane that contains $\ell$. Any three points are always coplanar (since three non-collinear points define a plane, and three collinear points lie on many common planes). Coplanarity becomes a meaningful test with four or more points.

Intersections describe where geometric objects meet:

• Two non-parallel lines in the same plane intersect at exactly one point.
• A line and a plane intersect at exactly one point, unless the line lies entirely in the plane.
• Two non-parallel planes intersect along a line (not a point).

Properties for geometric problem-solving

Working with points, lines, and planes:

1. Two points define a line. Three non-collinear points define a plane. Use these facts to determine whether a figure is uniquely defined.
2. Check collinearity by asking whether all points can sit on one line. Check coplanarity by asking whether all points can sit in one plane.
3. When two lines intersect, the result is a point. When two planes intersect, the result is a line.

Working with angle relationships:

1. Set up equations using the relationship. For complementary angles: $x + y = 90$. For supplementary angles: $x + y = 180$. For vertical angles: $x = y$.
2. Substitute known values or expressions and solve for the unknown.
3. For intersecting lines, once you know one angle you can find all four. Vertical angles are equal, and each pair of adjacent angles is supplementary.

Combining concepts:

• When a line intersects a plane, identify the single point of intersection and use it to find angle relationships at that point.
• When two planes intersect along a line, angles formed between the planes (called dihedral angles) can be analyzed using the same vertical and supplementary relationships.
• If you know a vertical angle formed by two intersecting planes measures 110°, the angle across from it is also 110°, and each adjacent angle measures 70°.