Question

Will give medal and fan.
Given that WZ/YZ=VZ/XZ, select the postulate or theorem that you can use to conclude that the triangles are similar.

A. ASA Similarity Postulate
B. SSS Similarity Theorem
C. AA Similarity Postulate
D. SAS Similarity Theorem

Answer 1

Okay, The hypothesis (given information) tells us that segment WZ and YZ are proportional,
and that sg. VZ and XZ are also proportional.
Meaning that we have actually two sides of the triangles proportional.
We have some choices here, we could prove that XY and WV are proportional and use SSS but it would be too complex, so let's discard that.

Maybe we could use AA but for that we need to be given some special conditions that actually are not present, so, AA discarded.

Perhaps finding two equal angles in order to use ASA, but for that we would have to go through unecessary process.

Since we have two of the sides of each triangle proportional to each other, we could find the angle between them and that is very simple to do:

(I would like you to tell me why they are congruent).
And with that, we have found the angle between the proportional sides, si that allows us to use SAS, allowing us to conclude that these two triangles (WZV and XYZ) are similar.

Answer 2

is the reason why the two triangles are similar is because thay have similar side or shape...o-o

Answer 3

That's incorrect.
In geometry, the diagram is a way to visualize, it does not prove anything.
The reason why these two triangle are similar is because the SAS (Side-Angle-Side) similarity was present.
In more simple words, say for example. How can you recognize a chair?.
You'd actually search for the conditions that makes a furniture, a chair.
four "foots" and a "seat" is enough information for me to conclude "this is a chair".

A little abstract example but it is the simplest I have. But is is exactly the same as what i just said.
But in this case there are three basic conditions that helps me recognize the similarity of two triangles:
\[1) ASA\]
If two triangles have two equal angles and the side, shared by the angles, proportional, then the two triangles MUST be similar.
\[2)LAL\]
If two triangles have two of their sides proportional and the angle between them congruent, then the triangles MUST be similar.
\[3)AAA\]
If two triangles have their three angles, congruent, then the two triangles MUST be similar.

So, as long as we prove one of these, we will conclude that any pair of triangles are similar.