Geometry question re: rays & angles

[Sebastian (a lady)]

So I'm doing some math with Artichoke (DS #3) using a third grade text from SMSG (teacher book here).

 

The section on angles and rays has the observation that angles are the union of two rays, not two line segments. (page 47 of the .pdf)

 

So he drew two rays that crossed, rather than having an endpoint in common as the vertex of the angle. Our question is does this form four angles (with opposite angles being similar) or are some of the resulting "angles" not really angles because they are formed by a subset that goes from the endpoint to another finite point (the intersection where the rays cross)?

 

Not sure if I'm phrasing this well enough to picture.

[Dana]

Mathematically, it is correct that an angle is defined as the union of two rays - definitely NOT line segments. (Confirmed in two of my texts. As a general rule, use the definition of the text you're using...as long as it's correct... some things are in flux - like the definition of natural numbers or whole numbers.)

 

However, both definitions I have have the angle emanating from the same point, so it would NOT make sense to have the two rays not having the same endpoint. Sounds like a poor illustration and bad example.

 

You'll talk about angle A from rays AB and AC (with A as the endpoint of each ray) or about angle BAC or CAB, but they do have to have the same endpoint.

[Sebastian (a lady)]

Mathematically, it is correct that an angle is defined as the union of two rays - definitely NOT line segments. (Confirmed in two of my texts. As a general rule, use the definition of the text you're using...as long as it's correct... some things are in flux - like the definition of natural numbers or whole numbers.)

 

However, both definitions I have have the angle emanating from the same point, so it would NOT make sense to have the two rays not having the same endpoint. Sounds like a poor illustration and bad example.

 

You'll talk about angle A from rays AB and AC (with A as the endpoint of each ray) or about angle BAC or CAB, but they do have to have the same endpoint.

 

Let me try to explain the picture he drew and asked about.

 

On a clock face, draw a ray from 6 to 12 with the endpoint at 6. Call this ray ST.

Draw another ray from 3 to 9 with the endpoint at 3. Call this ray AB.

The two rays cross at the center, X.

 

Do you have four angles (TXA, AXS, SXB and BXT)?

or

Do you only have one angle (BXT) because the other three angles are formed by finite portions of the rays (line segments SX and AX).

 

Would the situation change if AB and ST were lines instead of rays?

 

Are we overthinking this? (This was a picture DS drew, not an example from the text. But since the text made the point about angles being formed by rays, not line segments, I wasn't sure about the figure he'd drawn.)

[Jann in TX]

There is more to this than ONE simple definition.

 

I agree that the 'formal' definition is that an angle is formed when 2 rays intersect at a common vertex.

 

Now the 'tricky'-- logic stuff.

 

Every ray is part of a line.

 

Every segment is part of a line.

 

So if an angle is formed when 2 rays intersect-- you can 'extend' the rays into lines-- and now you have 4 angles!

 

My 'informal' class definition is " an angle is formed when 2 STRAIGHT THINGIES intersect. (straight thingies are lines, segments and rays-- in any combination).

 

It would totally mess up high school Geometry if you left the definition of an angle as ONLY the figure formed when 2 rays intersect at a common vertex.

 

Think about the FORMAL definition of a rectangle... a rectangle is a quadrilateral (closed figure made of 4 line segments) having all interior ANGLES measuring 90 degrees.

Rectangles are made from LINE SEGMENTS-- the interior ANGLES are made from segments...

 

Does this contradict the definition of an angle? NO--- the segments that form the rectangle can be EXTENDED to form rays/lines....

Edited November 30, 2011 by Jann in TX

[Dana]

Let me try to explain the picture he drew and asked about.

 

On a clock face, draw a ray from 6 to 12 with the endpoint at 6. Call this ray ST.

Draw another ray from 3 to 9 with the endpoint at 3. Call this ray AB.

The two rays cross at the center, X.

 

Do you have four angles (TXA, AXS, SXB and BXT)?

or

Do you only have one angle (BXT) because the other three angles are formed by finite portions of the rays (line segments SX and AX).

 

Would the situation change if AB and ST were lines instead of rays?

 

Are we overthinking this? (This was a picture DS drew, not an example from the text. But since the text made the point about angles being formed by rays, not line segments, I wasn't sure about the figure he'd drawn.)

 

Yes, you're overthinking this :)

 

A ray is defined by an endpoint and ANY other point on that ray.

In your example, you have ray XT and ray XB making up angle TXB.

 

You could make a case that this is the only angle in the illustration based on the definition... but I've never encountered this. Instead, you'd simply talk about rays XT and XB and wouldn't use points A and S since they're irrelevant for the angle. The angle is made up of rays XT and XB - NOT AB and ST because an angle has to have the rays at the common vertex.

 

A ray is defined based on two points with one as the endpoint. (points are undefined)

 

So in your example, if you wanted a different angle, you'd just use rays XA and XS (for instance) to get angle AXS. It's all in the definition. From the points you mentioned, you can construct any number of rays... but only some of them will form angles.

 

Based on definition, I'd argue then that with ray AB and ST, you don't have any angles since you don't have a common vertex.

[Teachin'Mine]

Yes. Two intersecting lines would form four angles. You could put four points - one on each section of the line a distance from the center - and then name them and the point where they intersect. That would give you the names of the four distinct angles - the intersecting point would be the middle letter for each of the angles. The angles opposite each other are called vertical angles and they are always equal. Two sets of vertical angles would be formed.

[letsplaymath]

There is more to this than ONE simple definition.

 

I agree that the 'formal' definition is that an angle is formed when 2 rays intersect at a common vertex.

 

Now the 'tricky'-- logic stuff.

 

Every ray is part of a line.

 

Every segment is part of a line.

 

So if an angle is formed when 2 rays intersect-- you can 'extend' the rays into lines-- and now you have 4 angles!

 

My 'informal' class definition is " an angle is formed when 2 STRAIGHT THINGIES intersect. (straight thingies are lines, segments and rays-- in any combination).

 

It would totally mess up high school Geometry if you left the definition of an angle as ONLY the figure formed when 2 rays intersect at a common vertex.

 

Think about the FORMAL definition of a rectangle... a rectangle is a quadrilateral (closed figure made of 4 line segments) having all interior ANGLES measuring 90 degrees.

Rectangles are made from LINE SEGMENTS-- the interior ANGLES are made from segments...

 

Does this contradict the definition of an angle? NO--- the segments that form the rectangle can be EXTENDED to form rays/lines....

 

:iagree: Never let a pedantic definition get in the way of common sense! In your son's drawing, you can see 4 angles because there are four angles. Otherwise, it would be impossible for anyone to ever have any angle, 'cuz none of us can really draw a ray---our paper isn't big enough! But we can imagine all sorts of things in geometry...

[creekland]

Two intersecting lines or segments or rays create the vertex of angles. We then know that the angle itself goes on and on and on in one direction - meaning whatever we have drawn on paper just gives us the measurement tools to see what size it is. Thus, whatever you had to begin with are now rays as far as the angle is concerned. (The angle, itself, will go from the vertex, or intersecting point, out forever.)

 

Note that you actually have two angles from two rays - the minor angle "inside" the rays and the major angle "outside" the rays. Most commonly we are talking about the minor angle, but either fit the definition. If one is supposed to actually use the major angle there will be words or a circular mark made near the vertex to let the student know.

 

If my student were told to draw an angle from two rays I would want to see the vertex of both rays at the same point to be certain they understood what we were talking about.

[barbarajon]

Yes, Two intersecting lines forming a 'X’ then there are 4 angles. Though it is made up of two rays; we assume 4 points to determine it. (a ray is named as AB so we assume A is start point & B is any point on a ray) so two rays ray AB & ray XY intersecting at point ‘P’ then there are 4 angles as angle APX, angle APY, angle BPX & angle BPY.

I must add, yes you are overthinking it.