Question

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

Explanation:

Step1: Recall isosceles triangle properties

In isosceles triangle \(ABC\) with vertex \(B\), \(AB = BC\), so base angles \(∠A = ∠C\).

Step2: Use triangle angle sum theorem

Sum of angles: \(∠A + ∠B + ∠C = 180^\circ\). Given \(∠B = 130^\circ\), so \(∠A + ∠C = 180^\circ - 130^\circ = 50^\circ\). Since \(∠A = ∠C\), each is \(25^\circ\).

Step3: Check options

• First option: \(∠A=15^\circ, ∠C=35^\circ\) (not equal, false)
• Second option: \(∠A + ∠B=155^\circ\) → \(25^\circ + 130^\circ=155^\circ\) (true)
• Third option: \(∠A + ∠C=60^\circ\) → \(25^\circ + 25^\circ=50^\circ≠60^\circ\) (false)
• Fourth option: \(∠A=20^\circ, ∠C=30^\circ\) (not equal, false)

Answer:

The second option (m∠A + m∠B = 155°)