Triangle Proof? Beginner

[joanposkitt]

Hola friends,
I'm new at proofs, and was working through some basic examples, and wondered if someone would give me some help?

For example:
How can I prove that m ∠UDB = m ∠TBD, given,
A, B, C, and D be mutually distinct, coplanar points. Suppose lines
AB, BC, CD, and DA are all mutually distinct. Consider line segments AB,
BC, CD, and DA, each of which lies on the line having the same name.
This forms a four-sided shape ABCD, with vertexes at points A, B, C, and
D, in that order proceeding counterclockwise.
Let T be a point on line AB, such that point B is between point A and point T.
Let U be a point on
line DC such that point D is between point U and point C. Suppose that,the conditions the following are true:
• Line segment BT has the same length as line segment UD
• Line segment UB has the same length as line segment DT.

Joan

 

[HallsofIvy]

I presume you have a picture (I had to draw one to make sense of this). You should be able to see from your picture that triangles TBD and BDU are congruent triangles, having all three side congruent. Side BT is congruent to side DU,side TD is congruent to side BU, by hypothesis, and, of course, side BD is in both triangles. What do you know about congruent triangles?

 

[joanposkitt]

This is what I put together - however, it seems logical but to me, it doesn't really seem like a proof.


I did a little sketch - and this is what I have:
1. as opposite sides are the same length - they need to be parallel to connect
2. as opposite sides are parallel - this tells use we have a parallelogram
3. as we have a parallelogram, opposite angles are equal
4. this means triangle ABC and BCD are inverse congruent
5. lengths DU and BT are the same
6. since points U and T lay on the same line as AB and CD, which are parallel to each other, the points BTDU form another parallelogram

 

[HallsofIvy]

This is what I put together - however, it seems logical but to me, it doesn't really seem like a proof.

View attachment 4929
I did a little sketch - and this is what I have:
1. as opposite sides are the same length - they need to be parallel to connect

What opposite side are the same length? AB and CD? AD and BC? That was NOT given.

2. as opposite sides are parallel - this tells use we have a parallelogram

What opposite sides are parallel?

3. as we have a parallelogram, opposite angles are equal

You are told only that "This forms a four-sided shape ABCD, with vertexes at points A, B, C, and D, in that order proceeding counterclockwise." You are NOT told that opposite sides are the same length nor that they are parallel.

4. this means triangle ABC and BCD are inverse congruent

Since ABCD is not a parallelogram, this not true- but irrelevant to triangles UBD and TBD.

5. lengths DU and BT are the same

Yes, you were given this. But you do not mention DU and BT again.

[quore]6. since points U and T lay on the same line as AB and CD, which are parallel to each other, the points BTDU form another parallelogram[/QUOTE]
Nor is this true.

Lastly, you were asked to prove that "m ∠UDB = m ∠TBD" but you have said nothing about those angles. I told you exactly how to prove it in my first response. What can you say about triangles ADU and BDT?