To determine which statement must be true for the isosceles triangle ABC given that it has a [tex]$130^{\circ}$[/tex] angle at vertex B, let's analyze each statement carefully.
We know that:
- In an isosceles triangle, two angles are equal.
- The sum of the interior angles in any triangle is always [tex]$180^{\circ}$[/tex].
Given:
- [tex]$\angle B = 130^{\circ}$[/tex].
Therefore:
- The sum of the other two angles must be [tex]$180^{\circ} - 130^{\circ} = 50^{circ}$[/tex].
- Since it's an isosceles triangle, the two equal angles (let's call them [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex]) must each be [tex]\( \frac{50^{\circ}}{2} = 25^{circ}\)[/tex].
Now, we'll review each statement:
1. [tex]$m_{\angle A} = 15^{\circ}$[/tex] and [tex]$m_{\angle C} = 35^{\circ}$[/tex]
- These angles sum to [tex]$15^{\circ} + 35^{\circ} = 50^{\circ}$[/tex] which matches our requirement that the other two angles sum to [tex]$50^{\circ}$[/tex].
- However, in an isosceles triangle, the two equal angles should be the same, but here [tex]\( \angle A \neq \angle C \)[/tex].
- Therefore, the statement is incorrect even though the sum fits.
2. [tex]$m_{\angle A} + m_{\angle B} = 155^{\circ}$[/tex]
- Here, [tex]\( \angle B = 130^{\circ} \)[/tex] and [tex]$m_{\angle A}$[/tex] is given by the nature of the isosceles triangle as [tex]$25^{\circ}$[/tex].
- [tex]$25^{\circ} + 130^{\circ} = 155^{\circ}$[/tex].
- This statement is factually correct.
3. [tex]$m_{\angle A} + m_{\angle C} = 60^{\circ}$[/tex]
- If both [tex]$\angle A$[/tex] and [tex]$\angle C$[/tex] are equal in an isosceles triangle and sum to [tex]$50^{\circ}$[/tex], their sum is not [tex]$60^{\circ}$[/tex].
- Therefore, this statement is incorrect.
4. [tex]$m_{\angle A} = 20^{\circ}$[/tex] and [tex]$m_{\angle C} = 30^{\circ}$[/tex]
- These angles also sum to [tex]$20^{\circ} + 30^{\circ} = 50^{\circ}$[/tex].
- However, [tex]\( \angle A \neq \angle C \)[/tex] whereas they should be equal in an isosceles triangle.
- Therefore, this statement is incorrect despite the sum fitting.
In summary:
- The correct statement that must be true, taking all the properties of the isosceles triangle into account, is:
[tex]\[ m_{\angle A} + m_{\angle B} = 155^{\circ} \][/tex]
So, the correct answer is:
[tex]\[ m_{\angle} A + m_{\angle B}=155^{\circ} \][/tex]
