Two rays are opposite rays, by definition, if 

(1) they have the same endpoint, and

(2) their union is a line.

The first letter in the name of a ray always refers to the endpoint of the ray. Therefore,  \(\displaystyle \overrightarrow{DA}\) has its endpoint at \(\displaystyle D\) and \(\displaystyle \overrightarrow{AD}\) has its endpoint at \(\displaystyle A\). The two rays are not opposite rays.

The first letter in the name of a ray refers to its endpoint; the second refers to the name of any other point on the ray. \(\displaystyle \overrightarrow{AB}\) and \(\displaystyle \overrightarrow{AC}\) are rays that have endpoint \(\displaystyle A\) and pass through \(\displaystyle B\) and \(\displaystyle C\), respectively. Those rays are indicated below in red and green, respectively.

As it turns out, the two rays are one and the same.

Two rays are opposite rays, by definition, if 

(1) they have the same endpoint, and

(2) their union is a line.

The first letter in the name of a ray refers to its endpoint; the second refers to the name of any other point on the ray. \(\displaystyle \overrightarrow{EC}\) and \(\displaystyle \overrightarrow{EF}\) both have endpoint \(\displaystyle E\), so the first criterion is met. \(\displaystyle \overrightarrow{EC}\) passes through point \(\displaystyle C\) and \(\displaystyle \overrightarrow{EF}\) passes through point \(\displaystyle F\); \(\displaystyle \overrightarrow{EC}\) and \(\displaystyle \overrightarrow{EF}\) are indicated below in green and red, respectively:

The union of the two rays is a line. Both criteria are met, so the rays are indeed opposite.

Three points are collinear if there is a single line that passes through all three. In the diagram below, it can be seen that the line that passes through \(\displaystyle A\) and \(\displaystyle B\) does not pass through \(\displaystyle E\).

Therefore, the three points are not collinear.

Three points are collinear if there is a single line that passes through all three. In the diagram below, it can be seen that such a line exists.

A plane can be named after any three points on the plane that are not on the same line. \(\displaystyle E\), \(\displaystyle D\), and \(\displaystyle F\) do not appear on the same line; for example, as can be seen below, the line that passes through \(\displaystyle E\) and \(\displaystyle F\) does not pass through \(\displaystyle D\).

Plane \(\displaystyle EDF\) is a valid name for the plane that includes this figure.

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

\(\displaystyle E\), \(\displaystyle D\), \(\displaystyle F\), and \(\displaystyle B\) are not a clockwise or counterclockwise ordering of the vertices, so Quadrilateral \(\displaystyle EDFB\) is not a valid name for the quadrilateral.

A line can be named after any two points it passes through. The line \(\displaystyle \overleftrightarrow{CF}\) is indicated in green below.

The line does not pass through \(\displaystyle D\), so \(\displaystyle D\) cannot be part of the name of the line. Specifically, \(\displaystyle \overleftrightarrow{DF}\) is not a valid name.

If \(\displaystyle l \parallel m\) and \(\displaystyle l \parallel n\) , then \(\displaystyle m \parallel n\), since two lines parallel to the same line are parallel to each other.

If \(\displaystyle m \angle1 + m \angle 2 = 180\), then \(\displaystyle m \parallel n\), since two same-side interior angles formed by transversal \(\displaystyle t\) are supplementary.

If \(\displaystyle \angle3 \cong \angle 4\), then \(\displaystyle m \parallel n\), since two alternate interior angles formed by transversal \(\displaystyle u\) are congruent.

However, \(\displaystyle \angle1 \cong \angle 5\) regardless of whether \(\displaystyle m\) and \(\displaystyle n\) are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is \(\displaystyle 180^{\circ }\) , each of the missing angles measures \(\displaystyle \frac{180-120}{2} =30^{\circ }\).