Geometric Proofs

Q

Quixie

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Nov 17, 2009
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Hi! I'm a little embarrassed to be here; I've never had such problems learning something before. However, I've spent hours attempting these and attempting them again and I can't seem to get it - or the real hold up, understand exactly what I'm doing wrong. I've already passed all my tests too, but the required proofs are holding me up and risking me not passing! So, any help you could provide to help me understand would be most appreciated.

Proof B
[attachment=0:23qtdvis]proofb.JPG[/attachment:23qtdvis]
This task is a geometric proof, and you will need to develop logical statements that are supported by reasons. Support each statement with a reason. Keep in mind:
1. The diagram is NOT drawn to scale.
2. There may be multiple ways to show your proof.
Most people prove this in 7–16 steps, so if you need to add additional rows to the table, feel free to do so. The number of steps to complete this task does not matter as much as making sure that all the STATEMENTS and REASONS you list are correct.

That is the problem.

My work is...

STATEMENTS REASONS
1. Line GF is the perpendicular bisector of line AC 1. Given
2. F is the midpoint of line AC 2. Bisector of a segment instersects a segment at its midpoint
3. Line FC is congruent to line AF 3. Midpoint of a segment divides a line into two equal segments
4. Line GF is congruent to line GF 4. reflexive property
5. Angle GFA and Angle GFC are right angles 5. A perpendicular bisector creates a 90 degree angle with the line it bisects.
6. Triangle GFC is congruent to triangle GFA 6 HL
7Angle C is congruent to angle A 7. corresponding parts of congruent triangles
8. Sides GC and GA are congruent 8. If two angles of a triangle are congruent, the sides opposite them are congruent.
9. Triangle AGC is isosceles 9. A triangle with two congruent sides is isosceles


The reason given for it being incorrect is :
"It is "Given" that the line segment GF is perpendicular to the line segment AC. At least one more statement and accompanying reason is needed to establish a pair of corresponding congruent angles.
However, it is not "Given" that either such line segment is "Given" to be the bisector of the other. As such, no midpoint is established and, in turn, no pair of congruent line segments are established at the base of the diagram.
Another set of corresponding congruent parts is represented by statement number four and its accompanying reason.
A third pair of corresponding congruent parts can be found by carefully considering the list of information next to the diagram."

So I see that the symbol I thought was perpendicular bisector was just perpendicular, so I tried again but got stuck half way through

1. Line GB is congruent to line GD 1. Given
2. Angle BGE is congruent to angle DGE 2. Given
3. Line GE is congruent with line GE 3. reflexive property
4. Triangle BGE is congruent with Triangle DGE 4. SAS
5. Line BE is congruent to line ED corresponding parts of congruent triangles
6. Point E is the midpoint of line BD 6. definition of midpoint


I see how to prove that BGD is isosceles and I know that triangle AGC is similar to triangle BGD and think my solution must come along those lines - but how do I prove that?

There are more problems I have wrong but I have an idea if I understand why I can't get this one I might be able to figure out the others. Thanks so much!!!
 

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Okay, but how do I prove they are similar in the first place? Obviously, angle BGD is congruent to angle AGC. That provides one angle.

Sorry if I'm being dense here!!!
 
how do I prove they are similar in the first place? Obviously, angle BGD is congruent to angle AGC. That provides one angle.
Click to expand...

Since it's given that GB and GD are congruent, then triangle BGD is an isosceles triangle by definition. Therefore, angle GBE and angle GDE are congruent. (Angles opposite the congruent sides of an isosceles triangle are congruent.) Does that help?
 
So that gives me three angles in the first triangle.. but nothing else to prove similarity...

Or do I prove that GE is perpendicular to BD with that... and thus BD and AC are parallel, and use the corresponding angle postulate to show congruency in the corresponding base angles of the respective triangles?
 
prove that GE is perpendicular to BD with that... and thus BD and AC are parallel, and use the corresponding angle postulate to show congruency in the corresponding base angles of the respective triangles?
Click to expand...

I think you just answered your won question. Well done. :)
 
Okay so a little more help yet... if you could tell me what I am still missing.. and preferably why so I don't just run into the same problem :)

Thanks!

Here's what I submitted and was sent back..

1. Line GB is congruent to line GD 1. Given
2. Angle BGE is congruent to angle DGE 2. Given
3. Line GE is congruent with line GE 3. reflexive property
4. Triangle BGE is congruent with Triangle DGE 4. SAS
5. Line BE is congruent to line ED corresponding parts of congruent triangles
6. Point E is the midpoint of line BD 6. definition of midpoint
7. Triangle GBD is isosceles 7. A triangle with two congruent sides is isosceles
8. Angle GBD is congruent to angle GDB 8. The base angles of isosceles triangles are congruent
9. Line GE is the perpendicular bisector of line BD 9. definition of perpendicular bisector
10. Line GF is perpendicular to line AC 10. Given
11. Line BD and line AC are parallel 11. Converse to Perpendicular Transversal Theorem
12. Angle GDB is congruent to angle GCA 12. Corresponding angle postulate
13. Angle GBD is congruent to angle GAC 13. Corresponding angle postulate
14. Triangles BGD and AGC are similar 14. AA theorom
15. Line GC is congruent to line GA Converse to Isosceles Triangle Theorem (Base Angle Theorem)
16. Triangle AGC is isosceles 17. definition of isosceles

and here's the reason given
"What is indexed as statement and reason number fourteen may be more effective if it featured the two angles that were meant to be established as congruent before what is currently listed a the reason for statement number fifteen can be used.

Otherwise, the sequence of statements in the current draft appear to lead to a proof of the featured theorem."

and
"The reason that may be most effective in accompanying statement number fourteen is likely to emerge when a property that links statement numbers twelve and thirteen is considered"

I don't quite get the reason given and what I need to do to improve it :)
 
8. Angle GBD is congruent to angle GDB 8. The base angles of isosceles triangles are congruent
.
.
.
12. Angle GDB is congruent to angle GCA 12. Corresponding angle postulate
13. Angle GBD is congruent to angle GAC 13. Corresponding angle postulate

"What is indexed as statement and reason number fourteen may be more effective if it featured the two angles that were meant to be established as congruent…”

"The reason that may be most effective in accompanying statement number fourteen is likely to emerge when a property that links statement numbers twelve and thirteen is considered"
Click to expand...

I think the comments suggest that a better choice for 14 might be

14. Angle GCA is congruent to angle GAC 14. Transitive Property.

Maybe just insert this as an extra line and push your current 14 into position 15.

Does that make sense to you?
 
Hi,

forgive me for not examining all your proof,
but how about this

Start with the 2 clues for the inner triangle.
Those 2 mean that the inner triangle is isosceles.
B and D are images via an axial symmetry through the line GEF
due to GB = GD in length and angles BGE and DGE being equal.
Therefore angle EBG equals angle EDG and so angles GED and GEB
are equal and sum to 180 degrees so they are both 90 degrees.
Since GF is perpendicular to AC, the inner triangle is a reduced version of the outer one.
The same shape in 2D is the same object in 3D at different distances from the observer.
Hence the outer triangle is isosceles
 
Given:BCIIDF
BAIIDE
Prove: angle 1 is congruent to angle 2[attachment=0:24rqtcpo]5.jpg[/attachment:24rqtcpo]
 
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