1.1 Basic geometric terms and concepts

Fundamental Geometry Concepts

Points, lines, planes, angles, and shapes are the building blocks of everything in geometry. Every proof, construction, and theorem you'll encounter this year traces back to these basic terms, so having precise definitions matters more than you might expect.

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Key Geometric Terms

A point represents an exact location in space. It has no size at all: no length, no width, no height. You represent it with a dot and label it with a capital letter, like point $A$.

A line is a straight path that extends infinitely in both directions. Any two distinct points determine exactly one line. You'll see it drawn with arrows on both ends and named using two points on it, like $\overleftrightarrow{AB}$. Don't confuse a line (infinite) with a line segment (finite, with two endpoints, written $\overline{AB}$) or a ray (one endpoint, extending infinitely in one direction, written $\overrightarrow{AB}$).

A plane is a flat, two-dimensional surface that extends infinitely in all directions. Three non-collinear points (points that don't all lie on the same line) determine exactly one plane. Planes are usually drawn as parallelograms and named with three points, like plane $ABC$, or sometimes with a single script letter like plane $\mathcal{P}$.

Collinear means points lie on the same line. Coplanar means points lie on the same plane. These terms come up constantly in proofs, so get comfortable with them now.

An angle is the figure formed by two rays that share a common endpoint called the vertex. Angles are measured in degrees, where a full rotation equals $360°$. You name an angle using three points (with the vertex in the middle), like $\angle ABC$, or by its vertex alone if there's no ambiguity, like $\angle B$. The main classifications are:

• Acute angle: greater than $0°$ and less than $90°$
• Right angle: exactly $90°$
• Obtuse angle: greater than $90°$ and less than $180°$
• Straight angle: exactly $180°$ (the two rays form a line)

A shape is a closed two-dimensional figure made up of points, lines, and/or curves. Triangles, circles, rectangles, and other polygons are all shapes.

Classification of Geometric Shapes

Triangles are polygons with three sides and three angles. The interior angles of any triangle always sum to $180°$. You can classify triangles by their sides:

• Equilateral: all three sides are equal (and all angles are $60°$)
• Isosceles: at least two sides are equal
• Scalene: no sides are equal

You can also classify triangles by their angles: acute (all angles less than $90°$), right (one angle equals $90°$), or obtuse (one angle greater than $90°$). A single triangle gets both labels, so you might describe one as a "right scalene triangle."

Quadrilaterals are polygons with four sides and four angles. The interior angles always sum to $360°$. This family includes parallelograms, rectangles, squares, rhombuses, and trapezoids, each with its own special properties you'll study in detail later.

A circle is the set of all points in a plane that are equidistant from a fixed center point. The radius is the distance from the center to any point on the circle, and the diameter is a segment through the center with both endpoints on the circle. The diameter is always twice the radius: $d = 2r$.

A polygon is any closed plane figure formed by three or more straight line segments, where each segment intersects exactly two others at their endpoints. Polygons are regular when all sides and all angles are equal (like an equilateral triangle or a square) and irregular when they're not.

Dimensions in Geometric Objects

Dimension refers to the number of independent directions needed to describe a geometric object.

• Zero dimensions (0D): A point. It has only position, with no length, width, or height.
• One dimension (1D): Lines, rays, and segments. These have length but no width or height.
• Two dimensions (2D): Plane figures like circles and polygons. These have length and width but no height.
• Three dimensions (3D): Solid figures like spheres, prisms, and polyhedra. These have length, width, and height.

A useful way to think about it: each new dimension adds a new direction you can move in. A point is fixed, a line lets you move forward/backward, a plane adds left/right, and a solid adds up/down.

Relationships Between Geometric Elements

Parallel lines are two lines in the same plane that never intersect. They maintain a constant distance between them. In notation, $\ell_1 \parallel \ell_2$ means the two lines are parallel. The "same plane" part of this definition matters: two lines that never intersect but are not in the same plane are called skew lines, which is a different relationship entirely.

Perpendicular lines are two lines that intersect at exactly $90°$. The symbol is $\perp$, so $\ell_1 \perp \ell_2$ means they're perpendicular.

Congruent figures have the same size and the same shape. All corresponding sides are equal in length, and all corresponding angles are equal in measure. The symbol is $\cong$.

Similar figures have the same shape but not necessarily the same size. Corresponding angles are still equal, but corresponding sides are proportional rather than equal. The symbol is $\sim$. For example, two triangles where one has sides of 3, 4, 5 and the other has sides of 6, 8, 10 are similar because each side of the larger triangle is exactly twice the corresponding side of the smaller one. The ratio between corresponding sides (here, $2:1$) is called the scale factor.