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If isosceles triangle ABC has a [tex]\(130^{\circ}\)[/tex] angle at vertex B, 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 :

GinnyAnsweridn
To determine which statement about isosceles triangle [tex]\(ABC\)[/tex] with [tex]\(m \angle B = 130^\circ\)[/tex] is true, we need to use the property that the sum of the interior angles in any triangle is always [tex]\(180^\circ\)[/tex].

Given:
- [tex]\( m \angle B = 130^\circ \)[/tex]

Since triangle [tex]\(ABC\)[/tex] is isosceles, it means two of its angles are equal. Let's denote these two equal angles as [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex]. Because it's isosceles, we have:
[tex]\[ m \angle A = m \angle C \][/tex]

The sum of the angles in any triangle is:
[tex]\[ m \angle A + m \angle B + m \angle C = 180^\circ \][/tex]

Substituting the given [tex]\( m \angle B \)[/tex] and using [tex]\( m \angle A = m \angle C \)[/tex], we get:
[tex]\[ m \angle A + 130^\circ + m \angle A = 180^\circ \][/tex]
This can be simplified to:
[tex]\[ 2m \angle A + 130^\circ = 180^\circ \][/tex]

Solve for [tex]\( m \angle A \)[/tex]:
[tex]\[ 2m \angle A = 180^\circ - 130^\circ \][/tex]
[tex]\[ 2m \angle A = 50^\circ \][/tex]
[tex]\[ m \angle A = \frac{50^\circ}{2} \][/tex]
[tex]\[ m \angle A = 25^\circ \][/tex]

Since [tex]\( m \angle A = m \angle C \)[/tex], it follows that:
[tex]\[ m \angle C = 25^\circ \][/tex]

Now, let's evaluate each statement to determine which one is true:

1. [tex]\(m \angle A = 15^\circ\)[/tex] and [tex]\(m \angle C = 35^\circ\)[/tex]:
We found that [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\( m \angle C = 25^\circ\)[/tex]. Therefore, this statement is false.

2. [tex]\(m \angle A + m \angle B = 155^\circ\)[/tex]:
Substituting the values we found:
[tex]\[ 25^\circ + 130^\circ = 155^\circ \][/tex]
This statement is true.

3. [tex]\(m \angle A + m \angle C = 60^\circ\)[/tex]:
Substituting the values we found:
[tex]\[ 25^\circ + 25^\circ = 50^\circ \][/tex]
This statement is false.

4. [tex]\(m \angle A = 20^\circ\)[/tex] and [tex]\(m \angle C = 30^\circ\)[/tex]:
We found that [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\( m \angle C = 25^\circ\)[/tex]. Therefore, this statement is false.

Hence, the statement that must be true is:

[tex]\[\boxed{m \angle A + m \angle B = 155^\circ}\][/tex]
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