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.

You are watching: What are opposite rays in geometry

Refer to the above figure.

True or false:

and
comprise a pair of opposite rays.

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.

Refer to the above figure.

True or false:

and
comprise a pair of opposite rays.

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.

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### Example Question #1544 : Basic Geometry

Refer to the above diagram.

True or false:

, and  are collinear points.

True

False

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.

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### Example Question #1545 : Basic Geometry

Refer to the above diagram.

True or false:

, , and
are collinear points.

True

False

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.

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### Example Question #1546 : Basic Geometry

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

.

False

True

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.

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### Example Question #1541 : Plane Geometry

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.

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### Example Question #1548 : Basic Geometry

Refer to the above diagram:

True or false:

may also called
.

False

True

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.

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### Example Question #1549 : Basic Geometry

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

?

and

Any of these statements can be used to prove that

.

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.

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### 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.

Explanation:

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

, each of the missing angles measures
.

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