To solve the problem, consider the properties of an isosceles triangle and the fact that the sum of the interior angles of any triangle is always [tex]\(180^\circ\)[/tex].
Given:
- Triangle ABC is isosceles with [tex]\( \angle B = 130^\circ \)[/tex].
- Since ABC is isosceles, the remaining two angles, [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex], are equal.
Let's denote the measures of [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] as [tex]\( x \)[/tex]:
1. The sum of all angles in a triangle is [tex]\(180^\circ\)[/tex]:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
2. Substitute the known values into the equation:
[tex]\[
x + 130^\circ + x = 180^\circ
\][/tex]
3. Combine like terms:
[tex]\[
2x + 130^\circ = 180^\circ
\][/tex]
4. Subtract [tex]\(130^\circ\)[/tex] from both sides:
[tex]\[
2x = 50^\circ
\][/tex]
5. Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 25^\circ
\][/tex]
Thus, both [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are [tex]\( 25^\circ \)[/tex].
Now, let's check which statement is true based on our findings:
1. [tex]\( \angle A = 15^\circ \)[/tex] and [tex]\( \angle C = 35^\circ \)[/tex]:
- This is false because [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are both [tex]\(25^\circ\)[/tex].
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 is true.
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 is false.
4. [tex]\( \angle A = 20^\circ \)[/tex] and [tex]\( \angle C = 30^\circ \)[/tex]:
- This is false because [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are both [tex]\(25^\circ\)[/tex].
Therefore, the correct statement is:
[tex]\[ \boxed{ \angle A + \angle B = 155^\circ } \][/tex]
So, the true statement is:
[tex]\[ m \angle A + m \angle B = 155^\circ \][/tex]
