Question

if isosceles triangle abc has a 130° angle at vertex b, which statement must be true?
( m angle a = 15 ^ { circ } ) and ( m angle c = 35 ^ { circ } )
( m angle a + m angle b = 155 ^ { circ } )
( m angle a + m angle c = 60 ^ { circ } )
( m angle a = 20 ^ { circ } ) and ( m angle c = 30 ^ { circ } )

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). For \(\triangle ABC\), \(m\angle A + m\angle B + m\angle C = 180^\circ\).

Step2: Analyze isosceles triangle

In an isosceles triangle, the two equal angles are the base angles. Since \(m\angle B = 130^\circ\) (an obtuse angle), it must be the vertex angle (the unequal one), so \(m\angle A = m\angle C\).

Step3: Calculate \(m\angle A + m\angle C\)

Substitute \(m\angle B = 130^\circ\) into the angle - sum formula: \(m\angle A + 130^\circ+m\angle C = 180^\circ\). Then \(m\angle A + m\angle C=180^\circ - 130^\circ = 50^\circ\)? Wait, no, let's re - evaluate the options.
Wait, let's check each option:

• Option 1: If \(m\angle A = 15^\circ\) and \(m\angle C = 35^\circ\), then \(m\angle A

eq m\angle C\), which contradicts the isosceles triangle property (since \(\angle B\) is the vertex angle, \(\angle A\) and \(\angle C\) should be equal).

• Option 2: We know that \(m\angle B = 130^\circ\), and from the angle - sum \(m\angle A+m\angle C = 50^\circ\), so \(m\angle A=25^\circ\) and \(m\angle C = 25^\circ\). Then \(m\angle A + m\angle B=25^\circ+130^\circ = 155^\circ\). Let's check the other options.
• Option 3: From \(m\angle A + m\angle B + m\angle C = 180^\circ\) and \(m\angle B = 130^\circ\), we get \(m\angle A + m\angle C=180 - 130=50^\circ

eq60^\circ\).

• Option 4: If \(m\angle A = 20^\circ\) and \(m\angle C = 30^\circ\), then \(m\angle A

eq m\angle C\), which contradicts the isosceles triangle property.

Wait, there is a mistake in my earlier calculation of \(m\angle A\) and \(m\angle C\). Since \(\angle B = 130^\circ\), then \(m\angle A=m\angle C=\frac{180 - 130}{2}=25^\circ\). Then \(m\angle A + m\angle B=25 + 130 = 155^\circ\), so option 2 is correct.

Answer:

\(m\angle A + m\angle B = 155^\circ\) (the option with this statement)