Answer :
Sure, let's go through the solution step-by-step.
1. Triangular Sum Property:
- In any triangle, the sum of the angles is always [tex]\(180^{\circ}\)[/tex].
2. Given Information:
- We are given an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\( \angle B = 130^{\circ} \)[/tex].
3. Finding Other Angles:
- In an isosceles triangle, two angles are equal. Since [tex]\( \angle B \)[/tex] is the vertex angle, [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are the base angles of the triangle and are equal.
- We need to find [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex].
4. Sum of Angles:
- According to the triangular sum property:
[tex]\[ \angle A + \angle B + \angle C = 180^{\circ} \][/tex]
- Substituting [tex]\( \angle B = 130^{\circ} \)[/tex]:
[tex]\[ \angle A + 130^{\circ} + \angle C = 180^{\circ} \][/tex]
- Simplifying, we get:
[tex]\[ \angle A + \angle C = 180^{\circ} - 130^{\circ} = 50^{\circ} \][/tex]
5. Verification of Statements:
- Statement 1: [tex]\(m \angle A = 15^{\circ}\)[/tex] and [tex]\(m \angle C = 35^{\circ}\)[/tex]
- If [tex]\(m \angle A\)[/tex] is [tex]\(15^{\circ}\)[/tex] and [tex]\(m \angle C\)[/tex] is [tex]\(35^{\circ}\)[/tex], then:
[tex]\[ 15^{\circ} + 35^{\circ} = 50^{\circ} \][/tex]
- So this statement is true, but does not need to be in alignment with a clear logical sequence based on the given information.
- Statement 2: [tex]\(m \angle A + m \angle B = 155^{\circ}\)[/tex]
- We already have [tex]\(\angle B = 130^{\circ}\)[/tex].
- To check this statement:
- If we assume [tex]\(\angle A = ?\)[/tex] degrees, then:
[tex]\[ ? + 130^{\circ} = 155^{\circ} \][/tex]
- Solving the equation:
[tex]\[ ? = 155^{\circ} - 130^{\circ} = 25^{\circ} \][/tex]
Results:
[tex]\[ m \angle A + m \angle B = 25^{\circ} + 130^{\circ} = 155^{\circ} But, the total will be = 180^\circ. \][/tex]
- Statement 3: [tex]\(m \angle A + m \angle C = 60^{\circ}\)[/tex]
- [tex]\(m \angle A + m \angle C = 60^{\circ}\)[/tex]
- Given the found result:
[tex]\[ \angle A + \angle C = 50^{\circ} \][/tex]
-So this is false.
- Statement 4: [tex]\(m \angle A = 20^{\circ}\)[/tex] and [tex]\(m \angle C = 30^{\circ}\)[/tex]
- If [tex]\(m \angle A\)[/tex] is [tex]\(20^{\circ}\)[/tex] and [tex]\(m \angle C\)[/tex] is [tex]\(30^{\circ}\)[/tex], then:
[tex]\[ m \angle A + m \angle C \leq 50^{\circ} \][/tex]
-So this is false as the triangle angles must be 180^\circ
Therefore, based on all the steps we have verified, the statement given true is:
Statement 2: [tex]\( m \angle A + m \angle B = 155^{\circ}\)[/tex] must be true.
1. Triangular Sum Property:
- In any triangle, the sum of the angles is always [tex]\(180^{\circ}\)[/tex].
2. Given Information:
- We are given an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\( \angle B = 130^{\circ} \)[/tex].
3. Finding Other Angles:
- In an isosceles triangle, two angles are equal. Since [tex]\( \angle B \)[/tex] is the vertex angle, [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are the base angles of the triangle and are equal.
- We need to find [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex].
4. Sum of Angles:
- According to the triangular sum property:
[tex]\[ \angle A + \angle B + \angle C = 180^{\circ} \][/tex]
- Substituting [tex]\( \angle B = 130^{\circ} \)[/tex]:
[tex]\[ \angle A + 130^{\circ} + \angle C = 180^{\circ} \][/tex]
- Simplifying, we get:
[tex]\[ \angle A + \angle C = 180^{\circ} - 130^{\circ} = 50^{\circ} \][/tex]
5. Verification of Statements:
- Statement 1: [tex]\(m \angle A = 15^{\circ}\)[/tex] and [tex]\(m \angle C = 35^{\circ}\)[/tex]
- If [tex]\(m \angle A\)[/tex] is [tex]\(15^{\circ}\)[/tex] and [tex]\(m \angle C\)[/tex] is [tex]\(35^{\circ}\)[/tex], then:
[tex]\[ 15^{\circ} + 35^{\circ} = 50^{\circ} \][/tex]
- So this statement is true, but does not need to be in alignment with a clear logical sequence based on the given information.
- Statement 2: [tex]\(m \angle A + m \angle B = 155^{\circ}\)[/tex]
- We already have [tex]\(\angle B = 130^{\circ}\)[/tex].
- To check this statement:
- If we assume [tex]\(\angle A = ?\)[/tex] degrees, then:
[tex]\[ ? + 130^{\circ} = 155^{\circ} \][/tex]
- Solving the equation:
[tex]\[ ? = 155^{\circ} - 130^{\circ} = 25^{\circ} \][/tex]
Results:
[tex]\[ m \angle A + m \angle B = 25^{\circ} + 130^{\circ} = 155^{\circ} But, the total will be = 180^\circ. \][/tex]
- Statement 3: [tex]\(m \angle A + m \angle C = 60^{\circ}\)[/tex]
- [tex]\(m \angle A + m \angle C = 60^{\circ}\)[/tex]
- Given the found result:
[tex]\[ \angle A + \angle C = 50^{\circ} \][/tex]
-So this is false.
- Statement 4: [tex]\(m \angle A = 20^{\circ}\)[/tex] and [tex]\(m \angle C = 30^{\circ}\)[/tex]
- If [tex]\(m \angle A\)[/tex] is [tex]\(20^{\circ}\)[/tex] and [tex]\(m \angle C\)[/tex] is [tex]\(30^{\circ}\)[/tex], then:
[tex]\[ m \angle A + m \angle C \leq 50^{\circ} \][/tex]
-So this is false as the triangle angles must be 180^\circ
Therefore, based on all the steps we have verified, the statement given true is:
Statement 2: [tex]\( m \angle A + m \angle B = 155^{\circ}\)[/tex] must be true.
