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Linear Pairs

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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? After completing this Concept, you'll be able to use the definition of linear pair to solve problems like this one.

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CK-12 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.

Example A

What is the measure of each angle?

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 &= 180^\circ\\10q & = 220\\q & = 22

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 B

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

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 C

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

Because linear pairs have to add up to 180^\circ , the other angle must be 180^\circ-150^\circ=30^\circ .

CK-12 Linear Pairs

Guided Practice

Use the diagram below. Note that \overline{NK} \perp \overleftrightarrow{IL} .

1. Name one linear pair of angles.

2. What is m\angle INL ?

3. What is m\angle LNK ?

4. If m\angle INJ = 63^\circ , find m\angle MNI .


1.  \angle MNL and \angle LNJ

2. 180^\circ

3. 90^\circ

4. 180^\circ - 63^\circ=117^\circ


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

  1. Linear pairs are congruent.
  2. Adjacent angles share a vertex.
  3. Adjacent angles overlap.
  4. Linear pairs are supplementary.
  5. Supplementary angles form linear pairs.

For exercise 6, find the value of x .

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

  1. 61^\circ
  2. 23^\circ
  3. 114^\circ
  4. 7^\circ
  5. 179^\circ
  6. z^\circ

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