Answer :
Let's consider the properties of an isosceles triangle and use the given information to determine the true statements.
### Step-by-Step Solution:
1. Known Information and Basics:
- An isosceles triangle has two equal angles.
- The sum of all interior angles in a triangle is always [tex]\(180^\circ\)[/tex].
- In triangle [tex]\(ABC\)[/tex], vertex [tex]\(B\)[/tex] is given to have an angle of [tex]\(130^\circ\)[/tex].
2. Identify Equal Angles:
- Since triangle [tex]\(ABC\)[/tex] is isosceles with [tex]\( \angle B = 130^\circ \)[/tex], the remaining two angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are equal (because the triangle is isosceles).
3. Calculate Sum of Remaining Two Angles:
- The sum of angles in any triangle is [tex]\(180^\circ\)[/tex].
- Thus, the combined measure of angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] would be:
[tex]\[ \angle A + \angle C = 180^\circ - \angle B = 180^\circ - 130^\circ = 50^\circ \][/tex]
4. Determine Measure of Each Equal Angle:
- Since [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are equal, each angle will be:
[tex]\[ \angle A = \angle C = \frac{\angle A + \angle C}{2} = \frac{50^\circ}{2} = 25^\circ \][/tex]
5. Verification Against Given Statements:
- Statement 1: [tex]\( m \angle A = 15^\circ \)[/tex] and [tex]\( m \angle C = 35^\circ \)[/tex]
[tex]\[ \text{This statement is incorrect because both angles } A \text{ and } C \text{ should be equal and calculated as } 25^\circ. \][/tex]
- Statement 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]
[tex]\[ \text{This statement is correct.} \][/tex]
- Statement 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]
[tex]\[ \text{This statement is incorrect.} \][/tex]
- Statement 4: [tex]\( m \angle A = 20^\circ \)[/tex] and [tex]\( m \angle C = 30^\circ \)[/tex]
[tex]\[ \text{This statement is incorrect because both angles } A \text{ and } C \text{ should be equal and calculated as } 25^\circ. \][/tex]
### Final Conclusion:
The correct statement is:
- [tex]\( m \angle A + m \angle B = 155^\circ \)[/tex]
### Step-by-Step Solution:
1. Known Information and Basics:
- An isosceles triangle has two equal angles.
- The sum of all interior angles in a triangle is always [tex]\(180^\circ\)[/tex].
- In triangle [tex]\(ABC\)[/tex], vertex [tex]\(B\)[/tex] is given to have an angle of [tex]\(130^\circ\)[/tex].
2. Identify Equal Angles:
- Since triangle [tex]\(ABC\)[/tex] is isosceles with [tex]\( \angle B = 130^\circ \)[/tex], the remaining two angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are equal (because the triangle is isosceles).
3. Calculate Sum of Remaining Two Angles:
- The sum of angles in any triangle is [tex]\(180^\circ\)[/tex].
- Thus, the combined measure of angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] would be:
[tex]\[ \angle A + \angle C = 180^\circ - \angle B = 180^\circ - 130^\circ = 50^\circ \][/tex]
4. Determine Measure of Each Equal Angle:
- Since [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are equal, each angle will be:
[tex]\[ \angle A = \angle C = \frac{\angle A + \angle C}{2} = \frac{50^\circ}{2} = 25^\circ \][/tex]
5. Verification Against Given Statements:
- Statement 1: [tex]\( m \angle A = 15^\circ \)[/tex] and [tex]\( m \angle C = 35^\circ \)[/tex]
[tex]\[ \text{This statement is incorrect because both angles } A \text{ and } C \text{ should be equal and calculated as } 25^\circ. \][/tex]
- Statement 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]
[tex]\[ \text{This statement is correct.} \][/tex]
- Statement 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]
[tex]\[ \text{This statement is incorrect.} \][/tex]
- Statement 4: [tex]\( m \angle A = 20^\circ \)[/tex] and [tex]\( m \angle C = 30^\circ \)[/tex]
[tex]\[ \text{This statement is incorrect because both angles } A \text{ and } C \text{ should be equal and calculated as } 25^\circ. \][/tex]
### Final Conclusion:
The correct statement is:
- [tex]\( m \angle A + m \angle B = 155^\circ \)[/tex]
