To address this question, we need to determine the measures of the angles \( \angle A \) and \( \angle C \) in an isosceles triangle \( ABC \) given that the angle at vertex \( B \) is \( 130^\circ \).
An isosceles triangle has two equal angles. In this triangle, since \( \angle B = 130^\circ \), the other two angles \( \angle A \) and \( \angle C \) are equal due to the isosceles property.
The sum of the angles in any triangle is always \( 180^\circ \). Therefore, we can write the equation for the sum of the angles in triangle \( ABC \):
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
Given that \( \angle B = 130^\circ \), we can substitute this value into the equation:
[tex]\[
\angle A + 130^\circ + \angle C = 180^\circ
\][/tex]
Since \( \angle A = \angle C \), we can set \( \angle A = \angle C = x \). Thus, the equation becomes:
[tex]\[
x + 130^\circ + x = 180^\circ
\][/tex]
Simplifying this equation, we get:
[tex]\[
2x + 130^\circ = 180^\circ
\][/tex]
To find \( x \), we isolate it by performing the following steps:
[tex]\[
2x = 180^\circ - 130^\circ
\][/tex]
[tex]\[
2x = 50^\circ
\][/tex]
[tex]\[
x = \frac{50^\circ}{2}
\][/tex]
[tex]\[
x = 25^\circ
\][/tex]
So, the measures of \( \angle A \) and \( \angle C \) are both \( 25^\circ \).
Next, we will evaluate each statement given in the question to see which one is true:
1. \( m \angle A = 15^\circ \) and \( m \angle C = 35^\circ \): This statement is incorrect because \( \angle A \) and \( \angle C \) are both \( 25^\circ \).
2. \( m \angle A + m \angle B = 155^\circ \):
Given \( m \angle A = 25^\circ \) and \( m \angle B = 130^\circ \):
[tex]\[
25^\circ + 130^\circ = 155^\circ
\][/tex]
This statement is true.
3. \( m \angle A + m \angle C = 60^\circ \):
Given \( m \angle A = 25^\circ \) and \( m \angle C = 25^\circ \):
[tex]\[
25^\circ + 25^\circ = 50^\circ
\][/tex]
This statement is false.
4. \( m \angle A = 20^\circ \) and \( m \angle C = 30^\circ \): This statement is incorrect because \( \angle A \) and \( \angle C \) are both \( 25^\circ \).
Therefore, the statement that must be true is:
[tex]\[
m \angle A + m \angle B = 155^\circ
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{m \angle A + m \angle B = 155^\circ}
\][/tex]
