Sure! Let's solve the problem step-by-step to determine which statement must be true for the given isosceles triangle [tex]\(ABC\)[/tex] with a [tex]\(130^\circ\)[/tex] angle at vertex [tex]\(B\)[/tex].
1. Understanding the Triangle:
- We are given an isosceles triangle [tex]\(ABC\)[/tex] with vertex angle [tex]\(B = 130^\circ\)[/tex].
- In an isosceles triangle, the two base angles are equal. So, angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are equal.
2. Sum of Angles in a Triangle:
- The sum of the angles in any triangle is always [tex]\(180^\circ\)[/tex].
- Therefore, we can write the equation:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
3. Set Up the Equation:
- Let [tex]\(\angle A = \angle C = x\)[/tex]. Since [tex]\( \angle B = 130^\circ\)[/tex], we substitute these values into our equation:
[tex]\[
x + 130^\circ + x = 180^\circ
\][/tex]
Simplifying this equation gives:
[tex]\[
2x + 130^\circ = 180^\circ
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- We subtract [tex]\(130^\circ\)[/tex] from both sides:
[tex]\[
2x = 180^\circ - 130^\circ
\][/tex]
[tex]\[
2x = 50^\circ
\][/tex]
- Next, we divide both sides by 2:
[tex]\[
x = 25^\circ
\][/tex]
- Thus, [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex].
5. Evaluate the Statements:
- [tex]\(\angle A = 15^\circ\)[/tex] and [tex]\(\angle C = 35^\circ\)[/tex]:
- This statement is false because we have [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex].
- [tex]\(\angle A + \angle B = 155^\circ\)[/tex]:
- This statement is true because [tex]\( \angle A + \angle B = 25^\circ + 130^\circ = 155^\circ \)[/tex].
- [tex]\(\angle A + \angle C = 60^\circ\)[/tex]:
- This statement is false because [tex]\( \angle A + \angle C = 25^\circ + 25^\circ = 50^\circ \)[/tex].
- [tex]\(\angle A = 20^\circ\)[/tex] and [tex]\(\angle C = 30^\circ\)[/tex]:
- This statement is false because we have [tex]\(\angle A = 25^\circ\)[/tex] and [tex]\(\angle C = 25^\circ\)[/tex].
Conclusion:
The only true statement from the choices provided is:
[tex]\[
\boxed{\angle A + \angle B = 155^\circ}
\][/tex]
