To determine which statement must be true, we need to thoroughly analyze the isosceles triangle [tex]$ABC$[/tex] where the vertex angle at [tex]$B$[/tex] is given as [tex]$130^{\circ}$[/tex].
1. Identifying the equal angles:
Since triangle [tex]$ABC$[/tex] is isosceles with [tex]$B$[/tex] as the vertex angle, the angles at [tex]$A$[/tex] and [tex]$C$[/tex] are equal. Let’s denote these equal angles by [tex]$x$[/tex].
2. Sum of the internal angles in a triangle:
The sum of all internal angles in any triangle is always [tex]$180^{\circ}$[/tex]. Therefore, we have:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
3. Substituting the given angle:
We know that angle [tex]$B$[/tex] is [tex]$130^{\circ}$[/tex], and both [tex]$\angle A$[/tex] and [tex]$\angle C$[/tex] are equal to [tex]$x$[/tex]. Thus, we can write:
[tex]\[
x + 130^\circ + x = 180^\circ
\][/tex]
Simplifying, we get:
[tex]\[
2x + 130^\circ = 180^\circ
\][/tex]
4. Solving for [tex]$x$[/tex]:
Isolating [tex]$x$[/tex], we subtract [tex]$130^\circ$[/tex] from both sides:
[tex]\[
2x = 50^\circ
\][/tex]
Dividing by 2, we find:
[tex]\[
x = 25^\circ
\][/tex]
Therefore, [tex]$\angle A = 25^\circ$[/tex] and [tex]$\angle C = 25^\circ$[/tex].
5. Checking the statements:
- First statement: [tex]\(m \angle A=15^{\circ} \text{ and } m \angle C=35^{\circ}\)[/tex]
This is incorrect because both [tex]$\angle A$[/tex] and [tex]$\angle C$[/tex] are [tex]$25^\circ$[/tex].
- Second statement: [tex]\( m_{\angle} A + m_{\angle} B = 155^{\circ} \)[/tex]
To verify: [tex]\(m \angle A + m \angle B = 25^\circ + 130^\circ = 155^\circ\)[/tex].
This statement is true.
- Third statement: [tex]\(m \angle A + m \angle C = 60^{\circ}\)[/tex]
To verify: [tex]\(m \angle A + m \angle C = 25^\circ + 25^\circ = 50^\circ\)[/tex].
This statement is incorrect.
- Fourth statement: [tex]\(m \angle A=20^{\circ} \text{ and } m \angle C=30^{\circ}\)[/tex]
This is incorrect because both [tex]$\angle A$[/tex] and [tex]$\angle C$[/tex] are [tex]$25^\circ$[/tex].
Therefore, the correct statement is:
[tex]\[
m_{\angle} A + m_{\angle} B = 155^{\circ}
\][/tex]
