By definition, two angles form a linear pair if and only if 

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

In the figure below,  and  are marked in green and red, respectively:

The two angles have the same vertex and share no interior points. However, they do not share a side. Therefore, they do not comprise a linear pair.

By definition, two angles form a linear pair if and only if 

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

In the figure below,  and  are marked in green and red, respectively:

The two angles have the same vertex and share no interior points. However, they do not share a side. Therefore, they do not comprise a linear pair.

By definition, two angles form a linear pair if and only if 

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

 is the angle with vertex ; its two sides are the rays  and , which have endpoint  and pass through  and , respectively.  has the same vertex; its two sides are the rays  and , which have endpoint  and pass through  and , respectively.  and  are indicated below in red and green, respectively:

The angles have the same vertex and they share a side. However, the interior of  is entirely contained in the interior of . The angles do not comprise a linear pair.

A quadrilateral is named after its four vertices in consecutive order, going clockwise or counterclockwise. Quadrilateral  is the figure in red, below:

, , , and  also name the four vertices in clockwise order. It follows that Quadrilateral  is another valid name for the figure.

A quadrilateral is named after its four vertices in consecutive order, going clockwise or counterclockwise. Quadrilateral  is the figure in red, below:

, , , and  name the four vertices in counterclockwise order. It follows that Quadrilateral  is another valid name for the figure.

 and  form a pair of vertical angles, and are consequently congruent whether or not it holds that . Therefore, whether the lines are parallel cannot be determined.

 and  are both inside the two lines  and , and they appear on opposite sides of transversal . They are thus alternate interior angles, by definition, and since they are congruent, then by the Converse of the Alternate Interior Angles Theorem, it follows that ,

 The letters in the name of a triangle name its vertices, so  refers to the triangle with vertices , , and . A triangle is named after its vertices in any order, so this triangle can also be called .

 and  form a linear pair of angles and are supplementary regardless of whether or not . 

 and  form a linear pair of angles, and are therefore supplementary; the same holds for  and . Angles that are supplementary to congruent angles are themselves congruent, so, since , it follows that .