# Angles: Complementary, Supplementary, Vertical, and Linear Pairs

You may want to review:

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

Two angles are *complementary*
if and only if
the sum of their measures is $\,90^{\circ}\,.$

Two angles are *supplementary*
if and only if
the sum of their measures is $\,180^{\circ}\,.$

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 $\,m\angle 1 + m\angle 2 = 180^{\circ}\,$

Rays that:

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

*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 $\,C\,$’.

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 $\,m\angle 1 + m\angle 2 = 180^{\circ}\,.$

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 $\,m\angle 1 = m\angle 2\ .$

Two lines are *parallel*
if and only if
they lie in the same plane
and do not intersect.

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 $\,m\,.$

*perpendicular*if and only if they form a right angle.

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

$7x-15= 2x+55$ | vertical angles have equal measures |

$5x-15=55$ | subtract $\,2x\,$ from both sides |

$5x=70$ | add $\,15\,$ to both sides |

$x=14$ | divide both sides by $\,5$ |