To solve this problem, we need to analyze an isosceles triangle ABC where the angle at vertex B is given as [tex]\(130^\circ\)[/tex].
1. Understanding Isosceles Triangle Properties:
- In an isosceles triangle, two sides are of equal length, and the angles opposite these sides are also equal.
- If the angle at vertex B is [tex]\(130^\circ\)[/tex], the base angles at vertices A and C (angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex]) must be equal.
2. Sum of Angles in a Triangle:
- The sum of all internal angles in any triangle is always [tex]\(180^\circ\)[/tex].
3. Calculation of Base Angles:
- Let angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] be denoted as [tex]\( x \)[/tex].
- Since the sum of all angles in the triangle is [tex]\(180^\circ\)[/tex], we can write:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
- Substitute the known values and the variable [tex]\( x \)[/tex]:
[tex]\[
x + 130^\circ + x = 180^\circ
\][/tex]
- Combine like terms:
[tex]\[
2x + 130^\circ = 180^\circ
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
2x = 180^\circ - 130^\circ = 50^\circ
\][/tex]
[tex]\[
x = 25^\circ
\][/tex]
- Therefore, [tex]\( \angle A = 25^\circ \)[/tex] and [tex]\( \angle C = 25^\circ \)[/tex].
4. Checking the Provided Statements:
- Statement 1: [tex]\( \angle A = 15^\circ \)[/tex] and [tex]\( \angle C = 35^\circ \)[/tex]
- This would violate our finding that [tex]\( \angle A = \angle C = 25^\circ \)[/tex]. Hence, this statement is false.
- Statement 2: [tex]\( \angle A + \angle B = 155^\circ \)[/tex]
- Calculate [tex]\( \angle A + \angle B \)[/tex]:
[tex]\[
25^\circ + 130^\circ = 155^\circ
\][/tex]
- This statement is true.
- Statement 3: [tex]\( \angle A + \angle C = 60^\circ \)[/tex]
- Calculate [tex]\( \angle A + \angle C \)[/tex]:
[tex]\[
25^\circ + 25^\circ = 50^\circ
\][/tex]
- This statement is false.
- Statement 4: [tex]\( \angle A = 20^\circ \)[/tex] and [tex]\( \angle C = 30^\circ \)[/tex]
- This would violate our result that [tex]\( \angle A = \angle C = 25^\circ \)[/tex]. Hence, this statement is false.
After analyzing the statements, the one that must be true is:
[tex]\[
\boxed{\angle A + \angle B = 155^\circ}
\][/tex]
