Question

question 6 of 16 (1 point) | question attempt: 1 of unlimited
partially correct
(b): your answer is incorrect.
for each part below, use the figure to fill in the blank.
if necessary, you may learn what the markings on a figure indicate.
(a) find ( a b ).
(b) find ( m angle f ).

Explanation:

Step1: Analyze triangle \(ABC\)

In \(\triangle ABC\), since \(\angle A=\angle B = \angle C=60^{\circ}\), \(\triangle ABC\) is an equilateral triangle. In an equilateral triangle, all sides are equal. Given \(BC = 6\), so \(AB=BC\).

Step2: Analyze triangle \(DEF\)

In \(\triangle DEF\), since \(DE = DF\) (marked with the same tick - marks), \(\triangle DEF\) is an isosceles triangle. The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle F=m\angle D\) (base angles of an isosceles triangle). We know that \(m\angle E = 40^{\circ}\). Using the formula \(m\angle E+m\angle D+m\angle F=180^{\circ}\), substituting \(m\angle D = m\angle F\), we get \(40^{\circ}+2m\angle F=180^{\circ}\). Then \(2m\angle F=180^{\circ}- 40^{\circ}=140^{\circ}\), and \(m\angle F=\frac{140^{\circ}}{2}=70^{\circ}\).

Answer:

(a) \(AB = 6\)
(b) \(m\angle F=70^{\circ}\)