Vertical Angles
Vertical angles are two non-adjacent angles formed by intersecting lines. In the picture below,
∠1 and ∠3
are vertical angles and
∠2 and ∠4
are vertical angles.
Notice that these angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.
Investigation: 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
? Find m∠2
.
What is the angle relationship between
∠1 and ∠4
? Find
m∠4
.
What is the angle relationship between
∠2 and ∠3
? Find
m∠3
.
Are any angles congruent? If so, write down the congruence statement.
From this investigation, hopefully you found out that
∠1≅∠3
and
∠2≅∠4
. This is our first theorem. That means it must be proven true in order to use it.
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.
From the picture above:∠1 and ∠2 are a linear pair∠2 and ∠3 are a linear pair∠3 and ∠4 are a linear pairAll of the equations=180∘, so set the first and second equation equal toeach other and the second and third.Cancel out the like termsm∠1+m∠2=180∘m∠2+m∠3=180∘m∠3+m∠4=180∘m∠1+m∠2=m∠2+m∠3ANDm∠2+m∠3=m∠3+m∠4m∠1=m∠3, m∠2=m∠4
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