Question

Geometry help please, 4 questions.

Answer 1

1)
3. AB=BD, BD=CE -> AB=CE
4. CE=AC
5. Transitive Property of Congruence (AB=CE, CE=AC -> AB=AC)
6. Definition of isosceles triangle

2) ABCD is a parallelogram: given
AD is parallel to BC: definition of parallelogram
<3=<2: given
(<1=<3, <3=<2) -> <1=<2: transitive property of congruence
DC biscects ADE: definition of biscect

3) lines parallel->lines do not intersect
lines do not intersect
conc: lines are parallel

not valid, because the conclusion does not follow from the premise
you would need another premise stating "if they do not intersect then they are parallel" or a premise stating that both directions of inference are allowed

simple alternative example so you can understand it better
the sun is round, so it can't be a cube
you look at something different than a cube
conclusion: you are staring at the sun
obviously this can't be always true as there are many other "not-cube" things and the sun is not the only possibility
while you may actually look at the sun it doesn't follow from the premise similar to the line example

4) is the same as 3) not valid, because the conclusion does not follow from the premise

Syllogism would be (a -> b, b -> c) -> (a->c)
a=you like a person
b=you know their birthday
c=you will party with them

if you like the person, you know their birthday and if you know their birthday, you will party with them
or in short: if you like a person, you will party with them

Syllogism is essentially removing an intermediary inference step