Question

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 ( mangle f ). ( mangle f=square^{circ} ) (b) find ( ab ). ( ab = square )

Explanation:

Part (a)

Step1: Identify triangle type

Triangle \( DEF \) has \( DE = FE \) (marked congruent), so it's isosceles with \( \angle D = \angle F \).

Step2: Sum of angles in triangle

Sum of angles in a triangle: \( 180^\circ \). Given \( \angle E = 130^\circ \), so \( \angle D + \angle F + 130^\circ = 180^\circ \).

Step3: Solve for \( \angle F \)

Since \( \angle D = \angle F \), let \( \angle F = x \). Then \( 2x + 130^\circ = 180^\circ \). Subtract \( 130^\circ \): \( 2x = 50^\circ \). Divide by 2: \( x = 25^\circ \).

Step1: Identify triangle type

Triangle \( ABC \) has all angles \( 60^\circ \), so it's equilateral (all sides equal).

Step2: Determine side length

Given \( BC = 12 \), in equilateral triangle, \( AB = BC \).

Answer:

\( 25 \)

Part (b)