1.15: Supplementary Angles

- Page ID
- 2130

Two angles that add to 180 degrees and when adjacent form a straight line.

## Linear Pairs

Two angles are adjacent if they have the same vertex, share a side, and do not overlap. \(\angle PSQ\) and \(\angle QSR\) are adjacent.

A linear pair is two angles that are adjacent and whose non-common sides form a straight line. If two angles are a linear pair, then they are supplementary (add up to \(180^{\circ}\)). \(\angle PSQ\) and \(\angle QSR\) are a linear pair.

What if you were given two angles of unknown size and were told they form a linear pair? How would you determine their angle measures?

For Example \(\PageIndex{1}\) and \(\PageIndex{2}\), use the diagram below. Note that \(\overline{NK} \perp \overleftrightarrow{IL}\).

Example \(\PageIndex{1}\)

Name one linear pair of angles.

Solution

\(\angle MNL\) and \(\angle LNJ\)

Example \(\PageIndex{2}\)

What is \(m \angle INL\)?

Solution

\(180^{\circ}\)

Example \(\PageIndex{3}\)

What is the measure of each angle?

Solution

These two angles are a linear pair, so they add up to \(180^{\circ}\).

\((7q−46)^{\circ}+(3q+6)^{\circ}=180^{\circ}\)

\(10q−40^{\circ}=220^{\circ}\)

\(10q=180^{\circ}\)

\(q=22^{\circ}\)

Plug in q to get the measure of each angle.

\(m \angle ABD=7(22^{\circ})−46^{\circ}=108^{\circ} \)

\(m \angle DBC=180^{\circ}−108^{\circ}=72^{\circ}\)

Example \(\PageIndex{4}\)

Are \(\angle CDA\) and \(\angle DAB\) a linear pair? Are they supplementary?

Solution

The two angles are not a linear pair because they do not have the same vertex. They are supplementary because they add up to \(180^{\circ}: 120^{\circ}+60^{\circ}=180^{\circ}\).

Example \(\PageIndex{5}\)

Find the measure of an angle that forms a linear pair with \(\angle MRS\) if \(m \angle MRS\) is \(150^{\circ}\).

Solution

Because linear pairs have to add up to \(180^{\circ}\), the other angle must be \(180^{\circ}−150^{\circ}=30^{\circ}\).

## Review

For 1-5, determine if the statement is true or false.

- Linear pairs are congruent.
- Adjacent angles share a vertex.
- Adjacent angles overlap.
- Linear pairs are supplementary.
- Supplementary angles form linear pairs.

For exercise 6, find the value of \(x\).

- Figure \(\PageIndex{6}\)

Find the measure of an angle that forms a linear pair with \(\angle MRS\) if \(m \angle MRS\) is:

- \(61^{\circ}\)
- \(23^{\circ}\)
- \(114^{\circ}\)
- \(7^{\circ}\)
- \(179^{\circ}\)
- \(z^{\circ}\)

## Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.

## Vocabulary

Term | Definition |

Adjacent Angles | Two angles are adjacent if they share aside and vertex. The word ‘adjacent’ means ‘beside’ or ‘next-to’. |

linear pair | Two angles form a linear pair if they are supplementary and adjacent. |

Diagram | A diagram is a drawing used to represent a mathematical problem. |

## Additional Resources

Interactive Element

Video: Complementary, Supplementary, and Vertical Angles

Activities: Supplementary Angles Discussion Questions

Study Aids: Angles Study Guide

Practice: Supplementary Angles

Real World: Supplementary Angles