# Basic Geometry : Plane Geometry

## Example Questions

### Example Question #11 : How To Find A Ray

Refer to the above figure.

True or false:  and  comprise a pair of opposite rays.

False

True

False

Explanation:

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,   has its endpoint at  and  has its endpoint at . The two rays are not opposite rays.

### Example Question #12 : How To Find A Ray

Refer to the above figure.

True or false:  and  comprise a pair of opposite rays.

False

True

False

Explanation:

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.  and  are rays that have endpoint  and pass through  and , respectively. Those rays are indicated below in red and green, respectively.

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

### Example Question #11 : Transformation

Refer to the above figure.

True or false:  and  comprise a pair of opposite rays.

True

False

True

Explanation:

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.  and  both have endpoint , so the first criterion is met.  passes through point  and  passes through point  and  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.

### Example Question #14 : How To Find A Ray

Refer to the above diagram.

True or false: , and  are collinear points.

False

True

False

Explanation:

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  and  does not pass through .

Therefore, the three points are not collinear.

### Example Question #12 : How To Find A Ray

Refer to the above diagram.

True or false: , and  are collinear points.

False

True

True

Explanation:

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.

### Example Question #11 : How To Find A Ray

True or false: The plane containing the above figure can be called Plane .

True

False

True

Explanation:

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

Plane  is a valid name for the plane that includes this figure.

### Example Question #12 : Lines

Refer to the above diagram.

True

False

False

Explanation:

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

, and  are not a clockwise or counterclockwise ordering of the vertices, so Quadrilateral  is not a valid name for the quadrilateral.

### Example Question #1541 : Basic Geometry

Refer to the above diagram:

True or false:  may also called .

True

False

False

Explanation:

A line can be named after any two points it passes through. The line  is indicated in green below.

The line does not pass through , so  cannot be part of the name of the line. Specifically,  is not a valid name.

### Example Question #1 : How To Find An Angle Of A Line

Examine the diagram. Which of these conditions does not prove that   ?

Any of these statements can be used to prove that .

and

Explanation:

If  and  , then , since two lines parallel to the same line are parallel to each other.

If , then , since two same-side interior angles formed by transversal  are supplementary.

If , then , since two alternate interior angles formed by transversal  are congruent.

However,  regardless of whether  and  are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

### Example Question #1 : How To Find An Angle Of A Line

An isosceles triangle has an interior angle that measures . What are the measures of its other two angles?

This triangle cannot exist.