Answer :
To solve this problem, let's go through the known facts and reasoning step-by-step.
### Step 1: Understanding the properties of the triangle
We have an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\( \angle B = 130^\circ \)[/tex]. In an isosceles triangle, two sides are equal, and thus two angles are equal. Let's denote the measures of these equal angles at vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as [tex]\(m_{\angle} A = x\)[/tex] and [tex]\(m_{\angle} C = x\)[/tex].
### Step 2: Sum of angles in a triangle
We know that the sum of the interior angles of any triangle is always [tex]\(180^\circ\)[/tex]. Therefore, we can write the equation:
[tex]\[ m_{\angle} A + m_{\angle} B + m_{\angle} C = 180^\circ \][/tex]
### Step 3: Substitute known values
We substitute [tex]\( m_{\angle} A = x \)[/tex], [tex]\( m_{\angle} B = 130^\circ \)[/tex], and [tex]\( m_{\angle} C = x \)[/tex] into the equation:
[tex]\[ x + 130^\circ + x = 180^\circ \][/tex]
[tex]\[ 2x + 130^\circ = 180^\circ \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ 2x = 180^\circ - 130^\circ \][/tex]
[tex]\[ 2x = 50^\circ \][/tex]
[tex]\[ x = 25^\circ \][/tex]
So, [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
### Step 5: Verify each statement
Now let’s verify each provided statement based on these angle measures:
1. [tex]\( m_{\angle} A = 15^\circ \)[/tex] and [tex]\( m_{\angle} C = 35^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
2. [tex]\( m_{\angle} A + m_{\angle} B = 155^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} B = 25^\circ + 130^\circ = 155^\circ \)[/tex]
- This statement is true.
3. [tex]\( m_{\angle} A + m_{\angle} C = 60^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} C = 25^\circ + 25^\circ = 50^\circ \)[/tex]
- This statement is false.
4. [tex]\( m_{\angle} A = 20^\circ \)[/tex] and [tex]\( m_{\angle} C = 30^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
### Conclusion
Among the given statements, the one that must be true is:
[tex]\[ m_{\angle} A + m_{\angle} B = 155^\circ \][/tex]
### Step 1: Understanding the properties of the triangle
We have an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\( \angle B = 130^\circ \)[/tex]. In an isosceles triangle, two sides are equal, and thus two angles are equal. Let's denote the measures of these equal angles at vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as [tex]\(m_{\angle} A = x\)[/tex] and [tex]\(m_{\angle} C = x\)[/tex].
### Step 2: Sum of angles in a triangle
We know that the sum of the interior angles of any triangle is always [tex]\(180^\circ\)[/tex]. Therefore, we can write the equation:
[tex]\[ m_{\angle} A + m_{\angle} B + m_{\angle} C = 180^\circ \][/tex]
### Step 3: Substitute known values
We substitute [tex]\( m_{\angle} A = x \)[/tex], [tex]\( m_{\angle} B = 130^\circ \)[/tex], and [tex]\( m_{\angle} C = x \)[/tex] into the equation:
[tex]\[ x + 130^\circ + x = 180^\circ \][/tex]
[tex]\[ 2x + 130^\circ = 180^\circ \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ 2x = 180^\circ - 130^\circ \][/tex]
[tex]\[ 2x = 50^\circ \][/tex]
[tex]\[ x = 25^\circ \][/tex]
So, [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
### Step 5: Verify each statement
Now let’s verify each provided statement based on these angle measures:
1. [tex]\( m_{\angle} A = 15^\circ \)[/tex] and [tex]\( m_{\angle} C = 35^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
2. [tex]\( m_{\angle} A + m_{\angle} B = 155^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} B = 25^\circ + 130^\circ = 155^\circ \)[/tex]
- This statement is true.
3. [tex]\( m_{\angle} A + m_{\angle} C = 60^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} C = 25^\circ + 25^\circ = 50^\circ \)[/tex]
- This statement is false.
4. [tex]\( m_{\angle} A = 20^\circ \)[/tex] and [tex]\( m_{\angle} C = 30^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].
### Conclusion
Among the given statements, the one that must be true is:
[tex]\[ m_{\angle} A + m_{\angle} B = 155^\circ \][/tex]
