In an isosceles triangle with a height 10 and a base 10, a square is inscribed with side x along the base of the triangle as shown above. What is the area of the square?

I’ll draw this as best I can:   Look OK? Now let me draw a few more segments in blue… See what’s going on there? All of the small triangles in the figure are the same! (You can prove this with triangle similarity/congruence rules easily enough—I won’t spend the time doing so here, though.) We (more…)

Regarding triangles question number 8 in your book…

I don’t know how to navigate your site yet, so please forgive me if you have already answered this question. Regarding question number 8 in your book, will you further explain why y=180-x is the same angle degree as the unmarked angle? I know y=2x because of geometry, but I do not understand how y can also equal x?

Triangles ABC and ABD share side AB…

Triangles ABC and ABD share side AB.
Triangle ABC has area Q and triangle ABD has area R. If AD is longer than AC and BD is longer than BC, which of the following could be true?
I-R> Q
II R=Q
III R < Q
I chose "I" only but the answer was E (all of them could be.) How can the second and third condition be true?
Thanks in advance!