1.15: Supplementary Angles

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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.

f-d_948f9829eba73297972a091b94dd4eb5a471d8e63bb9756af1d4e81f+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{1}\)

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.

f-d_23d540e9be461c6e22261aa41532c5d3b85f23f2ca346d29a61c4357+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{2}\)

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 \(\overline{NK} \perp \overleftrightarrow{IL}\).

f-d_ddbfde705abde0cdc132cdfcd782bec102f8b34c4b23a3ab4f4862ca+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{3}\)

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?

f-d_8bcec55351da12bf7020e7186bc467135e5d4365db11b290a133521d+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{4}\)

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?

f-d_3c72a69d58607c8bcd2ec43af9f9d1233f828398c238e97644cc7e0d+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{5}\)

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