If you know anything about proofs, I need help with this one badly!?

Question

Given: O bisects line AC
angle 3 is congruent to angle 4
Prove: line AC and line BD bisect eachother

Given: O bisects line AC
angle 3 is congruent to angle 4
Prove: line AC and line BD bisect eachother

I know that there isnt a picture but its the best that i can do....

it is a quadraleteral, line DC goes across the top, line AB goes across the bottom, line DB and line AC are met in the middle at O. angle DOA= angle 1, angle DAO= angle 3, angle COB= angle 2, angle OCB= angle 4

Answer 1

Not only do AC and BD bisect one another, it follows from the given information that ABCD is a parallelogram, not just any old quadrilateral.

I'm sufficiently old-fashioned to use "equals" for equal angles, instead of some words which mean exactly that but take longer to write, or the need to use a symbol not easily found in standard fonts.

Basically, the proof of what you want proceeds by showing that triangles OAD and OCB are indeed traditionally "conguent." This follows almost immediately since ang. 3 = ang. 4 means:

1. angle OAD = angle OCB.

In addition, since angles 1 and 2 are opposite one another (where two lines, AC and BD simply cross one another), they are equal, and thus:

2. angle AOD = angle COB.

(Note that the letter-order here is the same permutation on each side of the letters in equation 1. That's how things must be when proving congruence.)

But you're also given that O bisects the line AC. (I'm amazed that the fussy pedants still allow the concept of bisection; don't they realize that they could have told you that the line segment AO was congruent to the line segment CO? Boy, they really missed a chance for more high falutin' and obscurantist language there!) Therefore, again in those same two triangles:

3. OA = OC.

Thus, two angles and a corresponding side are equal, therefore:

4. Triangle OAD is congruent to Triangle OCB. (AAS)

Therefore OD = OB (corresponding sides), which means that:

5. LInes AC and BD bisect one another.

(You were already told that point O bisected AC; now it's been shown that O also bisects BD, so AC and BD bisect one another.)

Furthermore, though you weren't asked for this, because of point 1, lines AD and BC are parallel to one another (they are oppositely directed lines coming off the one traversal BD at the same angle). Also, it's now easy to show that triangles OBA and ODC are conguent (SAS), thus that angles ABO and CDO are equal, finally implying that AB and DC are parallel.

Thus each opposite pairs of lines in this mere "quadrilateral" are in fact parallel, and therefore the whole figure ABCD is a parallelogram.

Live long and prosper.

Answer 2

can you tell me where is line BD?