Question

triangles and congruence
a triangle abc has a 130° angle at vertex b, which statement must be true?
options:

• ( mangle a = 15^circ ) and ( mangle c = 35^circ )
• ( mangle a + mangle b = 180^circ )
• ( mangle a = 20^circ ) and ( mangle c = 20^circ )
• ( mangle a + mangle c = 50^circ )

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\). So \(m\angle A + m\angle B + m\angle C = 180^\circ\). Given \(m\angle B = 130^\circ\), we find \(m\angle A + m\angle C = 180^\circ - 130^\circ = 50^\circ\).

Step2: Analyze Each Option

• First option: \(m\angle A = 15^\circ\), \(m\angle C = 35^\circ\) gives \(15 + 35 = 50\), but this is one possible case, not "must be true".
• Second option: \(m\angle A + m\angle B = 180^\circ\) would mean \(m\angle C = 0^\circ\), impossible.
• Third option: \(m\angle A = 20^\circ\), \(m\angle C = 20^\circ\) gives \(20 + 20 = 40

eq50\), false.

• Fourth option: \(m\angle A + m\angle C = 50^\circ\) matches the angle sum result, so it must be true.

Answer:

The statement " \(m\angle A + m\angle C = 50^\circ\) " (the fourth option) must be true.