The sum of the lengths of any two sides of a triangle must be greater than the third side. if a triangle has one side that is 8 cm and a second side that is 1 cm less than twice the third side, what are the possible lengths for the second and third sides?

1st Side: 8

2nd Side: 2x - 1

3rd Side: x

(8) + (2x - 1) > x            (2x - 1) + x > 8                        (x) + (8) > 2x - 1

       2x + 7 > x                     3x - 1 > 8                                8  > x - 1

            2x > -7                         3x > 9                                9  > x

             x > -3.5 (disregard)       x > 3                                 x < 9

3rd Side: x  ⇒ (3 < x < 9)

2nd Side: 2x - 1 ⇒ 2(3) - 1 < x < 2(9) - 1   ⇒   5 < x < 17

the 3rd side must be between 3 and 9, the 2nd side must be between 5 and 17

∠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.