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Geometry Proof help

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Since I have had only half a course in a non-proofy geometry, and haven't yet sat down and learned this, I'm having trouble marking this one proof since dd did it so differently than the answer key. It's from Life of Fred, Chapter 5, Santa Clara. She took 2 more steps and a very different path that the textbook answer, and the last reason isn't exactly the same, even if it's the same step.


There is a drawing of an upside down isosceles triangle ABC. Each of the sides AC and BC has a perpendicular line segment (dotted, so in the problem they are called dotted lines) that meets the opposing corner (so that the one from side AC ends at angle ABC and the one from side BC ends at angle BAC.) The 2 dotted line segments are called AE and BC (with those line segments on top, of course).

If the dotted lines are perpendiculars and are also congruent, prove that triangle ABC is isosceles.

My daughter’s proof; the given is correct and simply shows what is stated in the problem. I’m using the less than symbol for the angle symbol since I can type that.


  • Given
  • m<BDA=90 degrees; m<AEB=90 degrees. Reason: def of perpendicular line segments
  • <BDA+m<AEB; Algebra (in LoF you’re allowed to simply say Algebra for a reason; not to worry, she also does Dressler where she has to be more specific)
  • <BDA is congruent to <AEB; Reason Definition of congruent angles
  • AB=AB; Algebra
  • line segment AB is congruent to line segment AB; reason Definition of congruent line segments
  • triangle ABD is congruent to triangle BAE; SAS (side angle side)
  • <ABE is congruent to <BAD; definition of congruent triangles

triangle ABC is isosceles; corollary of the ITT (isosceles triangle theorem

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She did not prove Angle-Side-Angle because the angles she has declared congruent are not formed by the sides she declared congruent...


Since the triangles are right triangles they can be declared congruent through 'hypotenuse-leg' (if LOF covers that reason)...not that it would help...


Proofs are so varried--there is NOT a standard form (similar maybe). The way one texts works formal proofs can greatly differ from the way a different text shows them. This is my first year to each Jacob's Geometry--some of his proofs are driving ME crazy! Luckily proofs are not on standardized tests--they can't be because they are not standard!!!

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I'm thanking God right now that we are doing Chalkdust..... because I can send proofs to DM for help!


I just spent that last week reviewing all proofs so far in dd's book so I can help her. Not putting the exact reasons would tend to make things unclear and impossible to follow to flow of the proof. I think she needs to put all reasons, more than just "Algebra." She did prove that the two triangles ABE and BAD are congruent, but I think it is SSA. Not sure if she proved the triangle is isosoles. I think there is more to prove, (we haven't gotten quite that far yet). Hopefully Jann in TX will respond!

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In order to prove the main triangle ABC is isosceles she needs to prove that 2 of the smaller triangles (each containing one of the ABC sides) are congruent--then use corresponding parts to prove the main one is isosceles.


The teacher in me just can't bring myself to solve the whole proof for her--but maybe these hints will help.




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Thanks, Jann. These hints will help. I'd rather you don't solve it for her, and she'll feel the same way (She doesn't want me to show her the proof in the book or even read one step.) She's been ill this week, and that may be why she's so far off. Or she may just need more help, learning, etc. Or both.

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