How can you prove a triangle is isosceles?

use the distance formula to see if at least two sides are congruent.

use the slope formula to see if any sides are perpendicular.

use the distance formula to see if all three sides are congruent.

use the slope formula to see if any sides are parallel.

The correct option is to Use the distance formula to see if at least two sides are congruent.  

Step-by-step explanation:

we know that isosceles triangle is the triangle in which two sides and two angles are equal. So, if we want to prove the triangle is isosceles we have to prove the two sides or two angles equal.

In the given options the first option is considered.

In first option only we prove at least two sides of triangle congruent by which results into two angles also congruent.

Hence, the correct option is to Use the distance formula to see if at least two sides are congruent.  

∠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.