Question

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

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\), i.e., \(m\angle A + m\angle B+m\angle C = 180^{\circ}\). Given \(m\angle B=130^{\circ}\).

Step2: Find the sum of \(\angle A\) and \(\angle C\)

Substitute \(m\angle B = 130^{\circ}\) into the angle - sum formula: \(m\angle A + m\angle C=180^{\circ}-m\angle B\).
So \(m\angle A + m\angle C=180 - 130=50^{\circ}\).
Since the triangle is isosceles, \(m\angle A=m\angle C\). Then \(m\angle A=m\angle C=\frac{180 - 130}{2}=25^{\circ}\).
Let's check each option:

• Option A: \(m\angle A = 15^{\circ}\) and \(m\angle C = 35^{\circ}\), \(15 + 35=50^{\circ}\) but \(m\angle A

eq m\angle C\) for an isosceles triangle, so this is wrong.

• Option B: \(m\angle A+m\angle B\). Since \(m\angle B = 130^{\circ}\) and \(m\angle A = 25^{\circ}\), \(m\angle A+m\angle B=25 + 130=155^{\circ}\).
• Option C: \(m\angle A + m\angle C=50^{\circ}

eq60^{\circ}\), so this is wrong.

• Option D: \(m\angle A = 20^{\circ}\) and \(m\angle C = 30^{\circ}\), \(20+30 = 50^{\circ}\) but \(m\angle A

eq m\angle C\) for an isosceles triangle, so this is wrong.

Answer:

B. \(m\angle A + m\angle B = 155^{\circ}\)