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Rule

Vertical Angles Theorem

Vertical angles are always congruent.

Two intersecting lines that form two pairs of vertical angles

Based on the characteristics of the diagram, the following relations hold true.

∠ 1 ≅ ∠ 3
∠ 2 ≅ ∠ 4

Proof

 Geometric Approach
Analyzing the diagram, it can be seen that ∠ 1 and ∠ 2 form a straight angle, so these are supplementary angles. Similarly, ∠ 2 and ∠ 3 are also supplementary angles.

Two intersecting lines that form two pairs of vertical angles

Therefore, by the Angle Addition Postulate, the sum of m∠ 1 and m∠ 2 is 180^(∘), and the sum of m∠ 2 and m∠ 3 is also 180^(∘). These facts can be used to express m∠ 2 in terms of m∠ 1 and in terms of m∠ 3.

Angle Addition Postulate Isolate m∠ 2
m∠ 1+m∠ 2 = 180^(∘) m∠ 2 = 180^(∘)-m∠ 1
m∠ 2+m∠ 3 = 180^(∘) m∠ 2 = 180^(∘)-m∠ 3

By the Transitive Property of Equality, the expressions representing m∠ 2 can be set equal to each other. m∠ 2= 180^(∘)-m∠ 1 m∠ 2= 180^(∘)-m∠ 3 ⇓ 180^(∘)-m∠ 1= 180^(∘)-m∠ 3 Then the equation can be simplified.

180^(∘)-m∠ 1=180^(∘)-m∠ 3
SubEqn

LHS-180^(∘)=RHS-180^(∘)

- m∠ 1=- m∠ 3
MultEqn

LHS * (- 1)=RHS* (- 1)

m∠ 1=m∠ 3

By the definition of congruent angles, this means that the vertical angles ∠ 1 and ∠ 3 are congruent angles. The same process can be used to prove ∠ 2 and ∠ 4 congruent.

Two-Column Proof

The previous proof can be summarized in the following two-column table.

Statements
Reasons
1.
l_1 and l_2 lines
1.
Given
2.
∠ 1 and ∠ 2 supplementary
2.
Definition of straight angle
3.
m∠ 1+m∠ 2=180^(∘)
3.
Definition of supplementary angles
4.
m∠ 2=180^(∘)-m∠ 1
4.
Subtraction Property of Equality
5.
∠ 2 and ∠ 3 supplementary
5.
Definition of straight angle
6.
m∠ 2+m∠ 3=180^(∘)
6.
Definition of supplementary angles
7.
m∠ 2=180^(∘)-m∠ 3
7.
Subtraction Property of Equality
8.
180^(∘)-m∠ 1=180^(∘)-m∠ 3
8.
Transitive Property of Equality
9.
m∠ 1=m∠ 3
9.
Subtraction and Multiplication Properties of Equality

Proof

 Using Transformations
Consider the points A, B, C, and D on each ray that starts at the point of intersection E of the two lines.

Two intersecting lines that form two pairs of vertical angles with some points

Suppose that points A and B are rotated 180^(∘) about point E.

Vertical Angles Proof Rotation About Intersection Point

The points A and B are mapped onto the points A' and B' after the rotation. This means that ∠ AEB is mapped onto ∠ A'EB'. Since rotations are a rigid motion, ∠ AEB and ∠ A'EB' are congruent angles. ∠ AEB ≅ ∠ A'EB' Since the point A' lies on EC and point B' lies on ED, ∠ A'EB' is congruent to ∠ CED. ∠ A'EB' ≅ ∠ CED By applying the Transitive Property of Congruence, it can be confirmed that ∠ AEB is congruent to ∠ CED. ∠ AEB ≅ ∠ A'EB' ∠ A'EB' ≅ ∠ CED ⇓ ∠ AEB ≅ ∠ CED

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