Proving Triangle Congruency

Mar 2020
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Is it possible to prove these triangles are congruent? I know they are equiangular but this is of no relevance in regards to triangle congruency. I would need to know that corresponding sides are congruent. Can anyone help with this? Thanks so much!
 

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skipjack

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Dec 2006
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The triangles are similar, but needn't be congruent, so congruence can't be proved.
 
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Mar 2020
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I got this question wrong on a test and would like to try to get the points back. My teachers argument was that segment CA and segment AC are congruent by the reflexive property of congruency. She then said that I could prove congruency by ASA theorem of congruency. How should go about proving this wrong? Or is it correct?
 

mathman

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May 2007
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The teacher is correct. There are three basic theorems for triangles to be congruent: all sides equal to corresponding (SSS). two angles and one side (ASA) and two sides and included angle (SAS).
 
Mar 2020
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I know about these three basic theorems. The diagram doesn’t say That segment AC is congruent to CA. Are the segments always congruent even though they are on difference triangles? Also wouldn’t this mean that angle A is always congruent to angle A of a different triangle. This is not the case in the diagram.
 

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skipjack

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The question shows triangles ABC and CHA, but doesn't state that point A in triangle ABC is the same point as point A in triangle CHA, or that point C in triangle ABC is the same point as point C in triangle CHA. I don't think it's reasonable to assert that this is implied by the way the labels A and C are used. If such an assertion is accepted, though, the triangles are congruent (ASA), as per the less confusing diagram below.
Congruency.PNG
 
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Mar 2020
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In this specific scenario it is evident that AC is congruent to line AC and it can then be determined that the triangles are congruent by ASA. I suppose that you are in this case correct that the assertion of the angle label are always corresponding. The problem that I was given was that the “reflexive” sides of the triangle were not shared sides of the triangle. I will attach the problem that I was given and maybe you can help. I understand the above explanation that you gave. Thank you very much by the way for your attention to this. The question was can it be proved that these triangles are congruent and if so write the congruency statement.
 

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skipjack

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In your attachment, two triangles are drawn separately, but two vertices of one triangle are given the same labels as two vertices of the other triangle, which makes it unclear whether the triangles are to be considered as separate or considered as sharing a side. If they are considered as separate triangles, they needn't be congruent. If they are considered as triangles that share a side, they must be congruent. If the problem was originally given to you without a diagram, you have drawn an inappropriate diagram.
 
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Mar 2020
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I agree. This was the exact diagram that was given on the test (i am looking at it now). I agree that it is unclear. The diagram was the only information that I was given and I was told to prove congruency and give congruency statement. It also said that if congruency cannot be determined to write not enough info. You mentioned, “If they are considered as separate triangles they needn’t be congruent,” and I am unsure if they are separate triangles or not. How can this be determine? Anyhow, under these conditions that were presented to me in the test problem, can it be determined that the triangles are congruent or is it unknown?
 

skipjack

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What was the exact wording of the problem, including any indication of how many marks were available, and any preamble for the entire test? Was the entire test about triangles?

If you give a congruency statement, such as $\small\triangle \text{ABC} \cong \triangle \text{CHA}$ (ASA), give the letters representing the vertices in the appropriate order (for example, don't state that triangles ABC, AHC are congruent).

If you are writing a detailed proof of congruency, state your assumptions and then choose your approach. A valid approach depends to some extent on how you have been taught to set out a geometrical proof.