Angles: Complementary, Supplementary, Vertical, and Linear Pairs

You may want to review:

• Segments, Rays, Angles
• More Terminology for Segments and Angles

For this web exercise, assume all angles are measured in degrees.

In particular:

• if $\,\angle 1\,$ and $\,\angle 2\,$ are complementary, then $\,m\angle 1 + m\angle 2 = 90^{\circ}\,$
• if $\,\angle 1\,$ and $\,\angle 2\,$ are supplementary, then $\,m\angle 1 + m\angle 2 = 180^{\circ}\,$

DEFINITION opposite rays

Rays that:

• share a common endpoint, and
• point in opposite directions

are called opposite rays.

Note:  If three points are on a line with $\,B\,$ between $\,A\,$ and $\,C\,,$ then $\,\overrightarrow{BA}\,$ and $\,\overrightarrow{BC}\,$ are opposite rays.

Recall that both $\,A{-}B{-}C\,$ and $\,C{-}B{-}A\,$ are notation for ‘$B\,$ is between $\,A\,$ and $\,C\,$’.

DEFINITION linear pair

Two angles are a linear pair if and only if

• they have a common side, and
• their other sides are opposite rays.

Note:  If $\,\angle 1\,$ and $\,\angle 2\,$ are a linear pair, then $\,m\angle 1 + m\angle 2 = 180^{\circ}\,.$

DEFINITION vertical angles

Two angles are vertical angles if and only if the sides of one angle are opposite rays to the sides of the other.

Note:   Vertical angles are the ‘opposite angles’ that are formed by two intersecting lines.

Note:  If $\,\angle 1\,$ and $\,\angle 2\,$ are vertical angles, then $\,m\angle 1 = m\angle 2\ .$

Parallel lines are studied in more detail in a future section, Parallel Lines.

The symbol ‘$\,\parallel\,$’ is used to denote parallel lines.

The sentence ‘$\,\ell\parallel m\,$’ is read as ‘$\,\ell \,$ is parallel to $\,m\,$’, and is true precisely when line $\,\ell\,$ is parallel to line $\,m\,.$

The symbol ‘$\,\perp\,$’ is used to denote perpendicular lines.

The sentence ‘$\,\ell\perp m\,$’ is read as ‘$\,\ell \,$ is perpendicular to $\,m\,$’, and is true precisely when line $\,\ell\,$ is perpendicular to line $\,m\,.$

Example