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If isosceles triangle [tex]$ABC$[/tex] has a [tex]$130^{\circ}$[/tex] angle at vertex [tex]$B$[/tex], which statement must be true?

A. [tex]$m \angle A=15^{\circ}$[/tex] and [tex]$m \angle C=35^{\circ}$[/tex]
B. [tex]$m_{\angle A}+m_{\angle B}=155^{\circ}$[/tex]
C. [tex]$m \angle A + m \angle C=60^{\circ}$[/tex]
D. [tex]$m \angle A=20^{\circ}$[/tex] and [tex]$m \angle C=30^{\circ}$[/tex]

Sagot :

GinnyAnswer
Let's analyze the given problem step-by-step to determine which statements must be true.

1. Understanding the Triangle:
- Given an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\(\angle B = 130^\circ\)[/tex].
- In an isosceles triangle, two angles are equal. Since [tex]\(\angle B\)[/tex] is different, the other two angles, [tex]\(\angle A\)[/tex] and [tex]\(\angle C\)[/tex], will be equal.

2. Sum of Angles in a Triangle:
- The sum of all angles in any triangle is always [tex]\(180^\circ\)[/tex].
- Therefore, [tex]\( \angle A + \angle B + \angle C = 180^\circ \)[/tex].
- Given [tex]\(\angle B = 130^\circ\)[/tex], we can write:
[tex]\[ \angle A + 130^\circ + \angle C = 180^\circ \][/tex]

3. Equal Angles in Isosceles Triangle:
- Since [tex]\(\angle A = \angle C\)[/tex], let [tex]\(\angle A = \angle C = x\)[/tex].
- So, the equation becomes:
[tex]\[ x + 130^\circ + x = 180^\circ \][/tex]
[tex]\[ 2x + 130^\circ = 180^\circ \][/tex]

4. Solving for [tex]\( x \)[/tex]:
- Subtract [tex]\(130^\circ\)[/tex] from both sides of the equation:
[tex]\[ 2x = 50^\circ \][/tex]
- Divide by 2:
[tex]\[ x = 25^\circ \][/tex]
- Therefore, [tex]\( \angle A = \angle C = 25^\circ \)[/tex].

5. Evaluating the Statements:
- Now we will check each of the provided statements to see which one is true:
1. [tex]\( m \angle A = 15^\circ \)[/tex] and [tex]\( m \angle C = 35^\circ\)[/tex]
- This statement is false because [tex]\( \angle A = \angle C = 25^\circ \)[/tex].
2. [tex]\( m_{\angle A} + m_{\angle B} = 155^\circ \)[/tex]
- This statement is true because [tex]\( \angle A + \angle B = 25^\circ + 130^\circ = 155^\circ \)[/tex].
3. [tex]\( m \angle A + m \angle C = 60^\circ \)[/tex]
- This statement is false because [tex]\( \angle A + \angle C = 25^\circ + 25^\circ = 50^\circ \)[/tex].
4. [tex]\( m \angle A = 20^\circ \)[/tex] and [tex]\( m \angle C = 30^\circ \)[/tex]
- This statement is false because [tex]\( \angle A = \angle C = 25^\circ \)[/tex].

Therefore, the statement that must be true 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]
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