GMAT Math assist » Problem-Solving concerns » Geometry » present » expertise rays

Refer come the above figure.

and i beg your pardon of the adhering to are the contrary rays?

Explanation:

Opposite rays begin at the exact same endpoint; their directions are opposite every other. Since

has endpoint
, us are searching for the ray that has endpoint
and walk in opposing direction - this beam is
.

You are watching: Which of the following are opposite rays

Explanation:

A ray deserve to be called with two letters, the first of which should be that is endpoint and also the 2nd of which have the right to be any type of other point on the ray.

has endpoint
. The only other marked point top top the ray is
, for this reason the correct an option is
.

A ray starts in ~ the suggest

and also has a hopeful slope of
. In i m sorry quadrants go some component of the beam lie?

Explanation:

A ray starts in ~ a solitary point and then proceeds in a straight line infinitely in some direction. The given ray starts in ~ the point

, i m sorry is in quadrant
, therefore we automatically know this need to be included in the answer. The ray has a hopeful slope of
, which way the next point is
, followed by
,
,
,
, and so on. By plotting this points we deserve to visualize that the ray starts in quadrant
, and also then continues infinitely into quadrant
, without ever crossing with any component of quadrant

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### Example question #1 : expertise Rays

In the above figure, which 2 rays have actually as your union

?

and

and

and

and

and

and

Explanation:

An angle, by definition, is the union of two rays through the very same endpoint. Their common endpoint is the vertex of the angle, i beg your pardon is always named by the center letter of a three-letter angle name. Therefore, us are in search of two rays v endpoint

. Because the an initial letter that the name of a beam is constantly its endpoint, we are trying to find two rays v
as the first letter in their names. This makes  and
the exactly choice.

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### Example concern #21 : present

In the over figure, offer the intersection of

and
.

Explanation:

can be watched to be completely contained in
- the is,
. The intersection of a collection and that subset is the subset, so the correct an answer is
.

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### Example concern #22 : present

In the over figure, offer the intersection of

and
.

Explanation:

The diagram listed below show

and
in red and green, respectively:

The intersection of

and
is the set of clues they have in common, which can be watched to it is in the section of the line v endpoints
and
. This figure is the heat segment
.

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### Example inquiry #23 : present

Which that the following could be another name for  in the over figure?

I)

II)

III)

I, II, and also III

I and also III only

I only

I and II only

II only

I only

Explanation:

A beam is called after that endpoint and also any other allude on the ray, in that order. Because

is the an initial letter in the name
,
is its endpoint, and any other name because that the ray must begin with
; this permits us come eliminate
. Also,
is eliminated, since a beam is called after two, not three, points.
is a correct choice, due to the fact that the first letter in this surname is endpoint
, and the 2nd letter surname a suggest on this ray.

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### Example inquiry #24 : present

In the above diagram, let

and
be the midpoints of
and
, respectively, and
and
be the midpoints of
and
, respectively. I m sorry of the adhering to is not a valid alternate name for
?

Each of the other options gives a valid alternative surname for

.

Explanation:

Below is the diagram through the points

, and
, as described, displayed in green. Also,
, the ray that has actually endpoint
and passes through
, is significant in red.

The ray likewise passes through

, and
, so
, and
are additionally valid names because that the ray. However, the beam does no pass through
, so
is not a valid name for the ray.
is the correct choice.

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### Example inquiry #25 : currently

Name a pair that opposite rays in the above figure.

and

and

and

and

and

and

Explanation:

Opposite rays space rays that have the very same endpoint are on the same line, but share only the endpoint.

Two name of opposite light ray must begin with the exact same letter, due to the fact that the first letter in the name of a ray is the endpoint. Also, the 2nd letters have to represent clues collinear v each other and also the endpoint, and also on opposite political parties of the shared endpoint. Of the 5 choices, only

and
together meet this criteria.

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### Example concern #26 : currently

In the over figure, i beg your pardon of the following gives a correct surname for the union of

and
?

Explanation:

and
are non-si rays through the exact same endpoint, so your union is one angle. The mutual endpoint that the rays, which below is
, is the crest of the developed angle, and is because of this the center letter of the surname of the angle; this letter is flanked by two letters representing one allude on every ray, i beg your pardon are
and
. The correct choice among those given is
.

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