To determine which statement is true for the given isosceles triangle [tex]\(ABC\)[/tex] with a [tex]\(130^\circ\)[/tex] angle at vertex [tex]\(B\)[/tex], follow these steps:
1. Understand the properties of an isosceles triangle:
In an isosceles triangle, two sides are equal, and the angles opposite those sides are also equal.
2. Determine the sum of angles in a triangle:
The sum of the interior angles in any triangle is always [tex]\(180^\circ\)[/tex].
3. Assign given information:
Given that [tex]\(m \angle B = 130^\circ\)[/tex], and this is an isosceles triangle with the equal angles at vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex].
4. Calculate the sum of the other two angles (Angle A and Angle C):
Since the sum of angles in any triangle is [tex]\(180^\circ\)[/tex],
[tex]\[
m \angle A + m \angle B + m \angle C = 180^\circ
\][/tex]
Substituting [tex]\(m \angle B = 130^\circ\)[/tex],
[tex]\[
m \angle A + m \angle C = 180^\circ - 130^\circ = 50^\circ
\][/tex]
5. Determine the individual measures of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are equal in an isosceles triangle,
[tex]\[
m \angle A = m \angle C = \frac{50^\circ}{2} = 25^\circ
\][/tex]
Now that we have determined [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\(m \angle C = 25^\circ\)[/tex], we can evaluate the given statements:
- [tex]\( m \angle A=15^\circ \)[/tex] and [tex]\( m \angle C=35^\circ \)[/tex]: False. We determined that both [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex] each measure [tex]\(25^\circ\)[/tex].
- [tex]\( m \angle A + m \angle B = 155^\circ\)[/tex]: True. [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\(m \angle B = 130^\circ\)[/tex], thus,
[tex]\[
25^\circ + 130^\circ = 155^\circ
\][/tex]
- [tex]\( m \angle A + m \angle C = 60^\circ\)[/tex]: False. We determined that [tex]\(m \angle A + m \angle C = 50^\circ\)[/tex].
- [tex]\( m \angle A=20^\circ \)[/tex] and [tex]\( m \angle C=30^\circ \)[/tex]: False. We determined that both [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex] each measure [tex]\(25^\circ\)[/tex].
Therefore, the correct statement is:
[tex]\( m \angle A + m \angle B = 155^\circ \)[/tex].
