To solve the problem, we need to determine which of the given statements must be true for an isosceles triangle ABC with a [tex]\(130^\circ\)[/tex] angle at vertex B.
1. Finding the angles at vertices A and C:
Since triangle ABC is isosceles with [tex]\(\angle B = 130^\circ\)[/tex], the other two angles, [tex]\(\angle A\)[/tex] and [tex]\(\angle C\)[/tex], must be equal. In any triangle, the sum of all angles is [tex]\(180^\circ\)[/tex].
Let [tex]\(\angle A\)[/tex] and [tex]\(\angle C\)[/tex] be denoted by [tex]\(A\)[/tex].
Then, the sum of the angles in triangle ABC is:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
Given [tex]\(\angle B = 130^\circ\)[/tex] and [tex]\(\angle A = \angle C = A\)[/tex], we have:
[tex]\[
A + 130^\circ + A = 180^\circ
\][/tex]
Simplifying this equation, we get:
[tex]\[
2A + 130^\circ = 180^\circ
\][/tex]
Solving for [tex]\(A\)[/tex], we subtract [tex]\(130^\circ\)[/tex] from both sides:
[tex]\[
2A = 50^\circ
\][/tex]
Dividing by 2, we find:
[tex]\[
A = 25^\circ
\][/tex]
Therefore, [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex].
2. Evaluating the statements:
- Statement 1: [tex]\(m \angle A = 15^\circ\)[/tex] and [tex]\(m \angle C = 35^\circ\)[/tex]
This statement is false because [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex], not [tex]\(15^\circ\)[/tex] and [tex]\(35^\circ\)[/tex].
- Statement 2: [tex]\(m \angle A + m \angle B = 155^\circ\)[/tex]
Calculating [tex]\(m \angle A + m \angle B\)[/tex]:
[tex]\[
25^\circ + 130^\circ = 155^\circ
\][/tex]
This statement is true.
- Statement 3: [tex]\(m \angle A + m \angle C = 60^\circ\)[/tex]
Calculating [tex]\(m \angle A + m \angle C\)[/tex]:
[tex]\[
25^\circ + 25^\circ = 50^\circ
\][/tex]
This statement is false.
- Statement 4: [tex]\(m \angle A = 20^\circ\)[/tex] and [tex]\(m \angle C = 30^\circ\)[/tex]
This statement is false because [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex], not [tex]\(20^\circ\)[/tex] and [tex]\(30^\circ\)[/tex].
The correct statement that must be true is:
[tex]\[
m \angle A + m \angle B = 155^\circ
\][/tex]
Hence, the true statement is:
[tex]\[
\boxed{m \angle A + m \angle B = 155^\circ}
\][/tex]
