Is the converse of parallel chords of a circle theorem true?

__Thm__: If two chords of a circle are parallel, then the arcs between them are congruent.

A tutoring student asked me if that was a two-way statement (e.g., If arcs between two chords are congruent, then the chords are parallel).

I could not disprove it on the spot, the book did not offer a proof of the theorem. *Is it true that if two chords are congruent, then the chords connecting their endpoints are parallel? *Prove it! (or disprove it).

If it is true, then why wouldn't the textbook have stated it as an iff statement (I read it very closely, and it was *not* iff).

Re: Is the converse of parallel chords of a circle theorem true?

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Originally Posted by

**mathDad** __Thm__: If two chords of a circle are parallel, then the arcs between them are congruent.

A tutoring student asked me if that was a two-way statement (e.g., If arcs between two chords are congruent, then the chords are parallel). I could not disprove it on the spot, the book did not offer a proof of the theorem. *Is it true that if two chords are congruent, then the chords connecting their endpoints are parallel? *Prove it! (or disprove it).

Of course it is not true. Any two chords of the same length subtend congruent arcs.

Re: Is the converse of parallel chords of a circle theorem true?

I don't mean the chords that correspond to the subtended arc. I mean the chords that are created by connecting the end point of one arc to the end point of the other arc. I don't have the graphical tools or skill to draw it, but basically, I'm talking about the converse of the given theorem (which is true).

Note that the given theorem does is not referring to the arc that corresponds with each chord, it is talking about the arcs which connect the endpoint of one chord to the endpoint of the other. So I'm asking about the converse.

Re: Is the converse of parallel chords of a circle theorem true?

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Originally Posted by

**Plato** Of course it is not true. Any two chords of the same length subtend congruent arcs.

Sorry Mr Plato..........The converse is correct . the chords connecting the adjacent end points of two congruent arcs of the same circle are parallel !!

Re: Is the converse of parallel chords of a circle theorem true?

Quote:

Originally Posted by

**mathDad** I don't mean the chords that correspond to the subtended arc. I mean the chords that are created by connecting the end point of one arc to the end point of the other arc. I don't have the graphical tools or skill to draw it, but basically, I'm talking about the converse of the given theorem (which is true).

Note that the given theorem does is not referring to the arc that corresponds with each chord, it is talking about the arcs which connect the endpoint of one chord to the endpoint of the other. So I'm asking about the converse.

Sorry but your phrase "arcs between them" confused me.

If A, B, C & D are points arranged clockwise on a circle in that order and $\displaystyle \overline{AB}\|\overline{CD}$ then the arcs $\displaystyle BC~\&~DA$ are congruent.

The converse is true also. Basically proved the same way for both.

The inscribed angles $\displaystyle \angle BAC \cong \angle ACD$.

One way from parallel lines. The converse from congruent arcs.

Re: Is the converse of parallel chords of a circle theorem true?