1.5: Angle Pairs
Learning Objectives
 Recognize complementary angles, supplementary angles, linear pairs and vertical angles.
 Apply the Linear Pair Postulate and the Vertical Angles Theorem.
Review Queue
Use the picture below to answer questions 13.
 Find
x .  Find
y .  Find
z .
Know What? A compass (as seen to the right) is used to determine the direction a person is traveling in. The angles between each direction are very important because they enable someone to be more specific and precise with their direction. In boating, captains use headings to determine which direction they are headed. A heading is the angle at which these compass lines intersect. So, a heading of
What headings have the same angle measure? What is the angle measure between each compass line?
Complementary Angles
Complementary: When two angles add up to
Complementary angles do not have to be congruent to each other, nor do they have to be next to each other.
Example 1: The two angles below are complementary.
Solution: Because the two angles are complementary, they add up to
Example 2: The two angles below are complementary. Find the measure of each angle.
Solution: Again, the two angles add up to
However, this is not what the question asks for. You need to plug
Supplementary Angles
Supplementary: When two angles add up to
Just like complementary angles, supplementary angles do not have to be congruent or touching.
Example 3: The two angles below are supplementary. If
Solution: Just like Examples 1 and 2, set up an equation. However, instead of equaling
Example 4: What is the measure of two congruent, supplementary angles?
Solution: Supplementary angles add up to
So, two congruent, supplementary angles are right angles, or
Linear Pairs
Adjacent Angles: Two angles that have the same vertex, share a side, and do not overlap.
Linear Pair: Two angles that are adjacent and whose noncommon sides form a straight line.
Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.
Example 5: Algebra Connection What is the value of each angle?
Solution: These two angles are a linear pair, so they are supplementary, or add up to
So, plug in
Example 6: Are
Solution: The two angles are not a linear pair because they do not have the same vertex. However, they are supplementary,
Vertical Angles
Vertical Angles: Two nonadjacent angles formed by intersecting lines.
Notice that these angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.
Investigation 15: Vertical Angle Relationships
 Draw two intersecting lines on your paper. Label the four angles created
∠1, ∠2, ∠3, and∠4 . See the picture above.  Take your protractor and find
m∠1 .  What is the angle relationship between
∠1 and∠2 ? Findm∠2 .  What is the angle relationship between
∠1 and∠4 ? Findm∠4 .  What is the angle relationship between
∠2 and∠3 ? Findm∠3 .  Are any angles congruent? If so, write down the congruence statement.
From this investigation, hopefully you found out that
Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.
We can prove the Vertical Angles Theorem using the same process we used above. However, let’s not use any specific values for the angles.
Recall that anytime the measures of two angles are equal, the angles are also congruent.
Example 7: Find
Solution:
Know What? Revisited The compass has several vertical angles and all of the smaller angles are
Review Questions
 Find the measure of an angle that is complementary to
∠ABC ifm∠ABC is
45∘ 
82∘ 
19∘ 
z∘

 Find the measure of an angle that is supplementary to
∠ABC ifm∠ABC is
45∘ 
118∘ 
32∘ 
x∘

Use the diagram below for exercises 37. Note that
 Name one pair of vertical angles.
 Name one linear pair of angles.
 Name two complementary angles.
 Name two supplementary angles.
 Given that
m∠IJN=63∘ , find:
m∠JNL 
m∠KNL 
m∠MNL 
m∠MNI

For 815, determine if the statement is ALWAYS true, SOMETIMES true or NEVER true.
 Vertical angles are congruent.
 Linear pairs are congruent.
 Complementary angles add up to
180∘ .  Supplementary angles add up to
180∘  Adjacent angles share a vertex.
 Adjacent angles overlap.
 Complementary angles are
45∘ .  The complement of
x∘ is(90−x)∘ .
For 1625, find the value of
 Find
x .  Find
y .
Find
Algebra Connection. Use factoring or the quadratic formula to solve for the variables.
Review Queue Answers

x+26=3x−8 34=2x 17=x 
(7y+6)∘=90∘ 7y=84∘ y=12∘ 
z+15=5z+9 6=4z 1.5=z
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