Proving AD is congruent to AE: How to Use Triangle Congruence Proofs

[Bad_Blood]

Homework Statement

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Given: <1 is congruent to <2, <3 is congruent to <4
Prove: AD is congruent to AE

The Attempt at a Solution

The given and then..
<5 is congruent to <6 by Vertical Angles Congruent
BF is congruent to CF because of the Isoceles Triangle Converse
<A is congruent to <A by Reflexive Property

 

[tiny-tim]

Welcome to PF!

The given and then..
<5 is congruent to <6 by Vertical Angles Congruent
BF is congruent to CF because of the Isoceles Triangle Converse
<A is congruent to <A by Reflexive Property

Hi Bad_Blood ! Welcome to PF!

hmm … two things …

First, you should only use the word "congruent" for triangles (or squares, or other shapes).

Lengths and angles aren't "congruent"… they're "equal"!

Second, in any proof whch involves congruence, you must specify the two triangles which you say are congruent, and then give the reason.

It's not enough just to give the reason, and assume that the examiner will know which triangles you mean!

Hint: which two triangles are you trying to prove are congruent?

 

[quantumdude]

I don't see a D or an E. Could you perhaps scan an image of the diagram and attach it to your next post?

 

[tiny-tim]

I think D and E are where CF and BF meet BA and CA respectively … that's consistent with the stated AD = AE.

 

[HallsofIvy]

tiny-tim, angles and line segments are "congruent". "equals" is reserved for "names" for the same thing. If x and y are angles (strictly speaking, are names for angles), then saying "x= y" means that x and y represent exactly the same angle, not just two angles with the same measure. "x is congruent to y" means that x and y have the same measure. Same for line segments.

Bad Blood, You can prove that triangles BFD and BFE (I am assuming that D is on AB and E on AC) are congruent by "ASA".

 

[tiny-tim]

tiny-tim, angles and line segments are "congruent". "equals" is reserved for "names" for the same thing. If x and y are angles (strictly speaking, are names for angles), then saying "x= y" means that x and y represent exactly the same angle, not just two angles with the same measure. "x is congruent to y" means that x and y have the same measure. Same for line segments.

Hi HallsofIvy!

That's bizarre!

So which symbol is used if one wants to write "Prove: AD is congruent to AE" as "Prove: AD [symbol] AE"?

I've always undestood the standard usage to be as in the following from wikipedia, at http://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles:

Two triangles are congruent if their corresponding sides and angles are equal.

wikipedia doesn't say "Two triangles are congruent if their corresponding sides and angles are congruent." … and nor would anyone I know.

(and if two lines had the same endpoints, I would say "they are not only equal, they are the same")

Is this a general American usage?

 

[HallsofIvy]

Certainly the textbooks I have seen have used the terminology "congruent" for segements and angles of the same measure. Thanks for the wikipedia reference. Personally, I would consider it not "precise" enough. Perhaps confusing "angle" with the measure of that angle. Interestingly, the wikipedia entry for congruence says "In geometry, two sets are called congruent if one can be transformed into the other by an isometry" which would apply as well to line segments and angles.