Circle Geometry Converse Theorems (1 Viewer)

[qqmore]

Just wondering if anyone knows or heard from any HSC markers that would we be marked down by not writing converse theorems in the HSC exams. I know my school accepts it when students do not write these converse theorems but I'm not* sure what other schools do.

For example, let's say we are trying to prove A,B,C,D is con-cyclic. We found out that the the interval AB subtends the same angle at two points C and D on the same side of AB. Could we simply just say "alternate angles in the same segments are = in a cyclic quadrilateral" rather than "A,B,C,D is con-cyclic because the interval AB subtends equal angles at points C and D on same side of AB". ? ------ Cuz its quite tedious writing all that....

Another example, lets say we proved a triangle ABC has a right angle at B. Can we just say "Since Angle ABC = 90 degrees, therefore points A,B,C are con-cyclic as angle in the semi circle is 90 degrees" rather than "A,B,C is con-cyclic because the circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex"


Hope you understand my question....

Thanks in advance....

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[tommykins]

i believe as long as you are able to express the logic/thought, it won't matter how.

Pretend points ABCD as concyclic.

We proved <abc = 180- <acb
.'. <abc + < acb = 180
.'. ABCD is concyclic (opp. <'s suppl.)

 

[Michaelmoo]

I dont do 4 unit yet, but in 3 unit circle trigonometry its fairly simple in regards with the reasoning. You can probably get away with not writing the converse, although it is quite easy to change the reasoning into a converse on the spot.

You sound like your memorising reasonings word for word. You dont have to.

The Board of Studies quotes: "reasoning must be logical and clear to comprehend"; they do not require complex terminology.

For Example:

Angle ABC = Angle BCD (opposite angles of a cyclic quadrilateral are supplementary, ABCD is a cyclic quad)

Converse: ABCD is a cyclic quad (quadrilateral with opposite angles supplementary is a cyclic quad, Angle ABC = Angle BCD)

Something along those will get you marks. Good Luck.

 

[qqmore]

Angle ABC = Angle BCD (opposite angles of a cyclic quadrilateral are supplementary, ABCD is a cyclic quad)

Converse: ABCD is a cyclic quad (quadrilateral with opposite angles supplementary is a cyclic quad, Angle ABC = Angle BCD)

Something along those will get you marks. Good Luck.

Yea that is case for the converse theorems for proving quadrilaterals (with opposite angles are supplementary) as cyclic quads which is pretty similar to the non-converse theorem.

But, for the converse of 'angles in the same segment are equal' would you need to write all "the interval of one side subtends equal angles at vertices of this same side"? which is dissimilar.... People would argue that you cant use the word 'segment' as you would be assuming it is a cyclic quadrilateral already.

This applies for the alternate segment theorems, angle in semi circle theorem, angle at centre of circle = twice at circumference etc... any that uses circle terminology ......