Straight Lines

• A line segment is the part of a line that consists of two points, known as endpoints, and all points between them.

  • The midpoint of a line segment is the point that separates the line segment into two congruent parts.
    • The midpoint of a line segment is unique
  • Congruent ($\cong$) line segments are two line segments that have the same length.

• The distance between two points $A$ and $B$ is the length of the line segment $\overline{AB}$ that joins the two points.

• Ray $AB$, denoted by $\overrightarrow{AB}$ , is the union of $\overline{AB}$ and all points $X$ on $\overleftrightarrow{AB}$ such that $B$ is between $A$ and $X$.

• Parallel lines $\parallel$ are lines that lie in the same plane but do not intersect.

• Perpendicular lines $\perp$ are two lines that meet to form congruent adjacent angles.

  • If two lines are perpendicular, then they meet to form right angles.

• A number of lines are concurrent if they have exactly one point in common.

• Through two distinct points, there is exactly one line.

• Ruler Postulate - The measure of any line segment is a unique positive number

• Segment-Addition Postulate - If $X$ is a point on $\overline{AB}$ and A-X-B, then $AX+XB+AB$.

• If two distinct lines intersect, they intersect at a point.

• Through three noncollinear points, there is exactly one plane

• Axiom 1 - Given a pair of straight lines ${\ell }_1$ and ${\ell }_2$, there are three possibilities:

  • The lines converge: ${\ell }_2={\ell }_1$.
  • The lines are parallel: ${\ell }_2\parallel {\ell }_1$.
  • The lines intersect at exactly one point.

• Axiom 3 - Given a line $\ell$ and a point $A$ that is not on the line, there exists $m$ unique line passing through the point $A$ and parallel to the straight line

• $({\ell }_1\parallel {\ell }_3)\wedge ({\ell }_2\parallel {\ell }_3)⟹({\ell }_1\parallel {\ell }_2)\vee {\ell }_1={\ell }_2$

• Given two parallel lines ${\ell }_2\parallel {\ell }_1$, any straight line intersected with ${\ell }_1$ also cuts ${\ell }_2$.

• Let ${\ell }_1$ be straight and let $P$ be a point. Then there is a single straight line passing through $P$ and perpendicular to $\ell$.