Question

1. find ( mangle b ) and ( ac ).

Explanation:

Step1: Determine the type of triangle

Since \(AB = BC=3m\), triangle \(ABC\) is isosceles. In an isosceles triangle, the base angles are equal.

Step2: Calculate \(m\angle B\)

The sum of angles in a triangle is \(180^{\circ}\). Let \(m\angle A=m\angle C = 60^{\circ}\). Then \(m\angle B=180^{\circ}-(m\angle A + m\angle C)\). Substitute \(m\angle A = 60^{\circ}\) and \(m\angle C=60^{\circ}\) into the formula: \(m\angle B=180^{\circ}-(60^{\circ}+ 60^{\circ})=60^{\circ}\).

Step3: Determine the type of triangle (again)

Since \(m\angle A=m\angle B = m\angle C=60^{\circ}\), triangle \(ABC\) is equilateral.

Step4: Find the length of \(AC\)

In an equilateral triangle, all sides are equal. Since \(AB = 3m\) and \(BC = 3m\), then \(AC=3m\).

Answer:

\(m\angle B = 60^{\circ}\) and \(AC = 3m\)