Geometry Proof - At it for Hours to No Avail

In summary, you can find the equation for the similarity of two triangles by drawing a right-angled triangle with the given angles and using the right-angle theorem. Next, you use the reflexive property to determine that the triangles are congruent.
  • #1
meiso
29
0

Homework Statement



Given:

Right triangle ABC with right angle A

segment AD is an altitude of the triangle
segment AE is an angle bisector of angle BAC
segment AF is the median of side BC

note: side BC has points D,E, and F on it in that order

EDIT: I've added a picture of the figure:
http://westchesterccb.com/proofpic.JPG [Broken]

Prove:

angle DAE is congruent to angle EAF



Homework Equations



N/A


The Attempt at a Solution



Got three similar triangles in ABD ~ DAC ~ DBA

Sorry for the lack of formatting. To anyone who would attempt this proof and possibly solve it, I'd greatly appreciate it.
 
Last edited by a moderator:
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  • #2
Welcome to PF!

Hi meiso! Welcome to PF! :smile:

Hint: that's the same as proving that angle BAD = angle FAC :wink:
 
  • #3
Yes, thank you for that hint. I had realized that and I still have not been able to solve it.
I also have the equations (either from givens or similar triangles and substitution):

angle ABC = angle DAE + angle EAF + angle FAC
angle BAD + angle DAE = angle EAF + angle FAC = 45 degrees
angle BAD = angle BCA

If I could somehow prove angle ABC = angle BAD + angle DAE + angle EAF,
then AF = BF, then BF = FC, and by the base angles theorem,
angle FAC = angle FCA, then by the transitive property, angle BAD = angle FAC, and the rest is easy from there. However, I can't seem to establish that link above.
 
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  • #4
meiso said:
Yes, thank you for that hint. I had realized that and I still have not been able to solve it.

ah … well, that's why it's so important for you to start by telliing us what you have tried

ok, new hint:

you have a right-angled triangle, so draw it in a circle :wink:
 
  • #5
Wow. Very elegant tiny-tim. Thank you. BC becomes the diameter of the circle, so BF, FC, and AF are all radii, so their measures are all equal, which is what I needed.

I was helping a high school student whom I tutor with this proof, and I never would have thought of using your method because he has not done anything like that in class. Thank you again, but if there is another method without using a circumscribed circle, I would love to know!
 
  • #6
meiso said:
… if there is another method without using a circumscribed circle, I would love to know!

ok … alternative hint:

draw FG parallel to AC :wink:
 
  • #7
Ok. So FG is drawn with G on segment BA, and intersects two sides of the triangle, so, being parallel to the base, it divides the other two sides proportionally.
Since BF=FC, BF/FC = 1, so BG/GA must equal 1 and and therefore BG=GA.

Following from that, I can use the SAS(BG=GA, two right angles, Reflexive GF) theorem to prove triangle BGF is congruent to triangle AGF. Triangle BGF is similar to triangle BAC (AA Theorem), so angle BFG = angle BCA. Angle AFG = angle FAC because they are alt. int. angles, but angle AFG also equals angle BCA (because angle BCA = angle BFG). By the transitive property (angle BCA = angle BAD already established), then, angle BAD = angle FAC!

This all led from your hint. Please tell me if any of my logic was incorrect (I know it's a lot to sift through) or if there is a shorter way to reach the conclusion.

And thanks again!
 
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  • #8
meiso said:
… Following from that, I can use the SAS(BG=GA, two right angles, Reflexive GF) theorem to prove triangle BGF is congruent to triangle AGF. …

Yes that's ok, but from there on, because I know the circle method, I'd concentrate on sides rather than angles:

FG perp BA, so (SAS) BF = AF, so CF = AF, so angle CAF = ACF :wink:
 

1. What is a geometry proof?

A geometry proof is a logical argument that uses deductive reasoning to show that a statement or theorem is true. It involves using known facts, definitions, and postulates to arrive at a conclusion.

2. How do you write a geometry proof?

To write a geometry proof, you must first state the given information and what you are trying to prove. Then, you must use deductive reasoning to make a series of logical steps, using previously proven theorems and postulates, to arrive at the conclusion. Make sure to clearly label each step and provide a reason for each step.

3. What are the different types of geometry proofs?

There are two main types of geometry proofs: two-column proofs and paragraph proofs. Two-column proofs involve writing each step of the proof in the left column and the corresponding reason in the right column. Paragraph proofs, on the other hand, are written in paragraph form and are more descriptive.

4. How can I improve my skills in writing geometry proofs?

Practice is key to improving your skills in writing geometry proofs. Make sure to thoroughly understand the definitions, postulates, and theorems before attempting a proof. Also, try breaking down the proof into smaller steps and use diagrams to help visualize the problem.

5. Why am I having trouble with geometry proofs?

Geometry proofs can be challenging because they require critical thinking and the ability to apply previously learned concepts. If you are having trouble, it may be helpful to review the definitions and theorems, seek assistance from a teacher or tutor, and practice more proofs to build your skills.

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