Which of the following is a negation for "There exists a real number x such that for all real numbers y, xy > y."1) There exists a real number x such that for all real numbers y, xy ≤ y.2) There exists a real number y such that for all real numbers x, xy ≤ y.3) There exists real numbers x and y such that xy ≤ y.4) For all real numbers x there exists a real number y such that xy ≤ y.5) For all real numbers y there exists a real number x such that xy ≤ y.

1) There exists a real number x such that for all real numbers y, xy ≤ y.

Step-by-step explanation:

Given the statement:

"There exists a real number x such that for all real numbers y, xy > y"

The negation of the statement is:

"There exists a real number x such that for all real numbers y, xy ≤ y"

The correct option is 1

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.

this answer is !