In depth investigation of vertical angles.

definition
Vertical angles are a pair of non-adjacent angles formed by the intersection of two straight line.

Vertical angles are located across from one another in the corners of the "X" formed by two straight lines.

In the diagram at the right, lines m and n are straight:
∠1 and ∠2 are vertical angles.
∠3 and ∠4 are vertical angles.

∠1 and ∠3 are NOT vertical angles.
vertucakangkes

theorem "Vertical Angle Theorem": Vertical angles are congruent.
vert4not
 vert4


Examples:
1. vert1
Given:
straight lines m and n

Find the number of degrees
in the indicated angles.
ANSWER:
The indicated angles are vertical angles.
5x - 6 = 3x + 12
2x = 18
x = 9
5(9) - 6 = 39º
3(9) + 12 = 39º

 
2. vert2
Given:
vertnot1
ANSWER:
m∠ACF=24º
m∠CAF=28º
m∠CAF=128º
m∠EFD=128º
m∠AFD=52º
m∠CFE=52º

 
3. vert3
Given:
vert3not
ANSWER:
The indicated angles are supplementary.
3x + 10 + 4x + 30 = 180
7x + 40 = 180
7x = 140
x = 20
m∠AEC=70º
m∠DEB=70º since it is vertical to ∠AEC.

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Proof of Vertical Angle Theorem - Using Transformations

The basis for this transformational proof will be a rotation of 180º about E.
vertanglesgiven1
 vert4

Proof:
• A rotation of 180º about point E will map point A onto EB such that A will lie on EB since we are dealing with straight segments. It will also map point C onto ED2 such that C will lie on ED2.
• The rotation will create ∠A'EC', which will be congruent to ∠BED since they are the same angles with the same sides (rays) and same vertex.
• Since ∠A'EC' is a 180º rotation of ∠AEC about E, ∠A'EC' congruent∠AEC since rotations are rigid transformations which preserve angle measure.
• ∠AECcongruent∠BED by the transitive property of congruence (or substitution).
• The same argument will apply to proving ∠AEDcongruent∠BEC.

 

Proof of Vertical Angle Theorem - Using Linear Pairs

For this proof, we will look at the linear pair relationships between adjacent angles about point E.
vertanglesgiven1
180prooflabeling2
vert5
Statements
Reasons
1. proofLP1
1. Given
2. ∠1, ∠2 form linear pair
    ∠3, ∠4 form linear pair
    ∠1, ∠4 form linear pair
2. A linear pair is a pair of adjacent angles that form a straight line.
3. ∠1, ∠2 are supplementary
    ∠3, ∠4 are supplementary
   ∠1, ∠4 are supplementary
3. Angles that form a linear pair are supplementary.
4.  m∠1 + m∠2 = 180
    m∠3 + m∠4 = 180
    m∠1 + m∠4 = 180
4. Supplementary angles are two angles the sum of whose measures is 180º.
5.  m∠1 + m∠2 = m∠1 + m∠4
    m∠1 + m∠4 = m∠3 + m∠4
5. Quantities equal to the same quantity are equal to each other. (Substitution or Transitive)
6.  m∠1 = m∠1;   m∠4 = m∠4
6. Reflexive property (quantity = itself).
7.  m∠2 = m∠4;    m∠3 = m∠1
7. If equals are subtracted from equals, the differences are equal.
8.   ddd
or proofvert889
8. Congruent angles are angles of equal measure.



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