Geometry Chapter 1: Points, Lines, Planes, Segments, and Distance

Comprehensive practice flashcards covering points, lines, planes, rays, segment addition, congruence, and coordinate distance formulas from Geometry Lesson 1.1 and 1.2 notes.

What is an undefined term in geometry, and which three terms belong to this category?

Undefined terms are words, usually readily understood, that are not formally defined. In geometry, point, line, and plane are considered undefined terms.

What is a defined term in geometry?

Defined terms are terms that can be defined using undefined terms or/and defined terms.

How is a mathematical definition defined?

A definition is an explanation that assigns properties to a mathematical object.

How is a point defined and represented in geometry?

A point is a specific location in space with no dimension, represented by a dot and named using an uppercase letter.

What is a line and how is it named?

A line is a connected straight path with no thickness that continues forever in both directions. It is named using two points on the line (such as line $AB$, line $BA$, $\overleftrightarrow{AB}$, or $\overleftrightarrow{BA}$) or a cursive letter at the end of the line.

What is a plane and how is it named?

A plane is a flat surface with no thickness that continues forever in ALL directions, represented by a parallelogram. It is named by a single script letter (e.g., plane $M$) or $3$ non-collinear points (e.g., Plane $ABC$).

What is a line segment and how is it named?

A line segment (or segment) is a portion of a line with $2$ endpoints and an infinite number of points between them. It is named by its endpoints with a bar symbol above them (such as segment $AB$, segment $BA$, $\overline{AB}$, or $\overline{BA}$).

What is a ray and how is it named?

A ray is a portion of a line that has $1$ endpoint, contains an infinite number of points, and continues forever in one direction. It is named by its endpoint followed by one other collinear point using a single arrow symbol (such as Ray $AB$ or $\vec{AB}$).

What are opposite rays?

Opposite rays are two rays that both start from a common point, extend in exactly opposite directions, and form a straight line.

What is the difference between collinear and noncollinear points?

Collinear points lie on the same line, whereas noncollinear points do not lie on the same line.

How many noncollinear points are needed to form a plane?

$3$ noncollinear points form a plane.

What is the difference between coplanar and noncoplanar points?

Coplanar points lie in the same plane, whereas noncoplanar points do not lie in the same plane.

How do parallel lines differ from skew lines?

Parallel lines are $2$ lines in a plane that do not meet, whereas skew lines are $2$ lines in different planes that do not meet.

What is the intersection of two intersecting planes?

Intersecting planes intersect at a line.

What is a postulate or axiom?

A postulate or axiom is a rule that is accepted without proof.

What is the Segment Addition Postulate (Betweenness of Points)?

If $B$ is between $A$ and $C$, then $AB + BC = AC$. Conversely, if $AB + BC = AC$, then $B$ is between $A$ and $C$.

What is the distinction between the notations $\overline{AB}$ and $AB$?

$\overline{AB}$ refers to the segment itself, whereas $AB$ refers to the length or distance of the segment.

If point $B$ is between $A$ and $C$, $AB = 7x - 5$, $BC = 4x - 6$, and $AC = 44$, what are the values of $x$ and $AB$?

Using $AB + BC = AC$, $(7x - 5) + (4x - 6) = 44 \Rightarrow 11x - 11 = 44 \Rightarrow 11x = 55 \Rightarrow x = 5$. Therefore, $AB = 7(5) - 5 = 30$.

What are congruent segments and how are they expressed symbolically versus as equal lengths?

Congruent segments are two segments with the same length. Equal lengths are written as $AB = CD$ ("is equal to"), while congruent segments are written as $\overline{AB} \cong \overline{CD}$ ("is congruent to").

How is the distance between two points $A$ and $B$ on a number line defined?

The distance between points $A$ and $B$ (or length of $AB$), written as $AB$, is the absolute value of the difference of their coordinates: $AB = |x_2 - x_1|$.

What is the Distance Formula for points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a coordinate plane?

$$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

What is the distance between $A(1, 2)$ and $B(3, 5)$ using the Distance Formula?

$$AB = \sqrt{(3 - 1)^2 + (5 - 2)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6$$