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From: Dan Christensen on 19 May 2010 01:07
Consider a line L, and degenerate triangle ABC in the Euclidean plane
such that:

1. A and B are distinct
2. A=C
3. A is on L
4. Line segment AB is perpendicular to L (so B is not on L)

Is there a general consensus about the degenerate triangle ABC having
(or not having) 2 right angles at A and C?

My thoughts: There is no angle BAC or BCA since A=C. And yet right
angles are formed between L and both line segments AB and CB. We can
also look at this triangle as a kind of limiting case: Two
perpendicular line segments, of equal length, each with an end point
on L, each on the same side of L, originally separate from one
another, but brought together to form a degenerate triangle. In this
case, right angles were defined before these two line segments were
brought together. Why not after?

Dan





From: Dan Christensen on 19 May 2010 01:12
(Correction)

Consider a line L, and degenerate triangle ABC in the Euclidean plane
such that:

1. A and B are distinct
2. A=C
3. A is on L
4. Line segment AB is perpendicular to L (so B is not on L)

Is there a general consensus about the degenerate triangle ABC having
(or not having) 2 right angles at A and C?

My thoughts: There is no angle BAC or BCA since A=C. And yet right
angles are formed between L and both line segments AB and CB. We can
also look at this triangle as a kind of limiting case: Two line
segments of equal length, each with an end point on L, each
perpendicular to L, each on the same side of L, each originally
separate from one another, but brought together to form a degenerate
triangle. In this case, right angles were defined before these two
line segments were brought together. Why not after?

Dan
From: Henry on 19 May 2010 04:13
On 19 May, 06:12, Dan Christensen <Dan_Christen...(a)sympatico.ca>
wrote:
> (Correction)
>
> Consider a line L, and degenerate triangle ABC in the Euclidean plane
> such that:
>
> 1. A and B are distinct
> 2. A=C
> 3. A is on L
> 4. Line segment AB is perpendicular to L (so B is not on L)
>
> Is there a general consensus about the degenerate triangle ABC having
> (or not having) 2 right angles at A and C?
>
> My thoughts: There is no angle BAC or BCA since A=C. And yet right
> angles are formed between L and both line segments AB and CB. We can
> also look at this triangle as a kind of limiting case: Two line
> segments of equal length, each with an end point on L, each
> perpendicular to L, each on the same side of L, each originally
> separate from one another, but brought together to form a degenerate
> triangle. In this case, right angles were defined before these two
> line segments were brought together. Why not after?
>
> Dan

The notional angles BAC or BCA can take on any values which sum to pi
(or 180 degrees) in the limit of the following case:

A and B fixed, a ray (half line) R from A at an arbitrary angle a to
AB, C a point on R, so angle BAC=a. In the limit, as C approaches A
along R, angle ABC approaches 0 and angle BCA approaches pi-a.
From: Dan Christensen on 19 May 2010 11:11
On May 19, 4:13 am, Henry <s...(a)btinternet.com> wrote:
> On 19 May, 06:12, Dan Christensen <Dan_Christen...(a)sympatico.ca>
> wrote:
>
>
>
>
>
> > (Correction)
>
> > Consider a line L, and degenerate triangle ABC in the Euclidean plane
> > such that:
>
> > 1. A and B are distinct
> > 2. A=C
> > 3. A is on L
> > 4. Line segment AB is perpendicular to L (so B is not on L)
>
> > Is there a general consensus about the degenerate triangle ABC having
> > (or not having) 2 right angles at A and C?
>
> > My thoughts: There is no angle BAC or BCA since A=C. And yet right
> > angles are formed between L and both line segments AB and CB. We can
> > also look at this triangle as a kind of limiting case: Two line
> > segments of equal length, each with an end point on L, each
> > perpendicular to L, each on the same side of L, each originally
> > separate from one another, but brought together to form a degenerate
> > triangle. In this case, right angles were defined before these two
> > line segments were brought together. Why not after?
>
> > Dan
>
> The notional angles BAC or BCA can take on any values which sum to pi
> (or 180 degrees) in the limit of the following case:
>
> A and B fixed, a ray (half line) R from A at an arbitrary angle a to
> AB, C a point on R, so angle BAC=a.  In the limit, as C approaches A
> along R, angle ABC approaches 0 and angle BCA approaches pi-a.

OK, my limiting case argument doesn't stand up. (A relief, actually!)
Thanks for clearing that up.

This issue came up yesterday when a student asked me: Can a triangle
have 2 right angles. My first response was, no, the two sides would be
parallel and never intersect (in the Euclidean plane). Then I began to
wonder if there wasn't a degenerate case with a zero-length base. (A
degenerate triangle is one with collinear vertices.) It appears there
isn't. Triangles (degenerate or otherwise) must be have 3 distinct
vertices. This is not usually made explicit in standard textbook
definitions. Shouldn't it be?

Dan
From: gudi on 19 May 2010 15:32
On May 19, 8:11 pm, Dan Christensen <Dan_Christen...(a)sympatico.ca>
wrote:
> On May 19, 4:13 am, Henry <s...(a)btinternet.com> wrote:
>
>
>
>
>
> > On 19 May, 06:12, Dan Christensen <Dan_Christen...(a)sympatico.ca>
> > wrote:
>
> > > (Correction)
>
> > > Consider a line L, and degenerate triangle ABC in the Euclidean plane
> > > such that:
>
> > > 1. A and B are distinct
> > > 2. A=C
> > > 3. A is on L
> > > 4. Line segment AB is perpendicular to L (so B is not on L)
>
> > > Is there a general consensus about the degenerate triangle ABC having
> > > (or not having) 2 right angles at A and C?
>
> > > My thoughts: There is no angle BAC or BCA since A=C. And yet right
> > > angles are formed between L and both line segments AB and CB. We can
> > > also look at this triangle as a kind of limiting case: Two line
> > > segments of equal length, each with an end point on L, each
> > > perpendicular to L, each on the same side of L, each originally
> > > separate from one another, but brought together to form a degenerate
> > > triangle. In this case, right angles were defined before these two
> > > line segments were brought together. Why not after?
>
> > > Dan
>
> > The notional angles BAC or BCA can take on any values which sum to pi
> > (or 180 degrees) in the limit of the following case:
>
> > A and B fixed, a ray (half line) R from A at an arbitrary angle a to
> > AB, C a point on R, so angle BAC=a.  In the limit, as C approaches A
> > along R, angle ABC approaches 0 and angle BCA approaches pi-a.
>
> OK, my limiting case argument doesn't stand up. (A relief, actually!)
> Thanks for clearing that up.
>
> This issue came up yesterday when a student asked me: Can a triangle
> have 2 right angles. My first response was, no, the two sides would be
> parallel and never intersect (in the Euclidean plane). Then I began to
> wonder if there wasn't a degenerate case with a zero-length base. (A
> degenerate triangle is one with collinear vertices.) It appears there
> isn't. Triangles (degenerate or otherwise) must be have 3 distinct
> vertices. This is not usually made explicit in standard textbook
> definitions. Shouldn't it be?
>
> Dan

A pair of parallel straight lines cut by a transversal produces two
such triangles.In 3D, two coincident longitudes for example,make up
such a degenerate case of two curved sides in spherical trig.

Narasimham
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