To determine which statement about isosceles triangle [tex]\(ABC\)[/tex] with [tex]\(m \angle B = 130^\circ\)[/tex] is true, we need to use the property that the sum of the interior angles in any triangle is always [tex]\(180^\circ\)[/tex].
Given:
- [tex]\( m \angle B = 130^\circ \)[/tex]
Since triangle [tex]\(ABC\)[/tex] is isosceles, it means two of its angles are equal. Let's denote these two equal angles as [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex]. Because it's isosceles, we have:
[tex]\[ m \angle A = m \angle C \][/tex]
The sum of the angles in any triangle is:
[tex]\[ m \angle A + m \angle B + m \angle C = 180^\circ \][/tex]
Substituting the given [tex]\( m \angle B \)[/tex] and using [tex]\( m \angle A = m \angle C \)[/tex], we get:
[tex]\[ m \angle A + 130^\circ + m \angle A = 180^\circ \][/tex]
This can be simplified to:
[tex]\[ 2m \angle A + 130^\circ = 180^\circ \][/tex]
Solve for [tex]\( m \angle A \)[/tex]:
[tex]\[ 2m \angle A = 180^\circ - 130^\circ \][/tex]
[tex]\[ 2m \angle A = 50^\circ \][/tex]
[tex]\[ m \angle A = \frac{50^\circ}{2} \][/tex]
[tex]\[ m \angle A = 25^\circ \][/tex]
Since [tex]\( m \angle A = m \angle C \)[/tex], it follows that:
[tex]\[ m \angle C = 25^\circ \][/tex]
Now, let's evaluate each statement to determine which one is true:
1. [tex]\(m \angle A = 15^\circ\)[/tex] and [tex]\(m \angle C = 35^\circ\)[/tex]:
We found that [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\( m \angle C = 25^\circ\)[/tex]. Therefore, this statement is false.
2. [tex]\(m \angle A + m \angle B = 155^\circ\)[/tex]:
Substituting the values we found:
[tex]\[ 25^\circ + 130^\circ = 155^\circ \][/tex]
This statement is true.
3. [tex]\(m \angle A + m \angle C = 60^\circ\)[/tex]:
Substituting the values we found:
[tex]\[ 25^\circ + 25^\circ = 50^\circ \][/tex]
This statement is false.
4. [tex]\(m \angle A = 20^\circ\)[/tex] and [tex]\(m \angle C = 30^\circ\)[/tex]:
We found that [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\( m \angle C = 25^\circ\)[/tex]. Therefore, this statement is false.
Hence, the statement that must be true is:
[tex]\[\boxed{m \angle A + m \angle B = 155^\circ}\][/tex]
