Question

In the diagram below of parallelogram ABCD, diagonal BED and EF

are drawn, EF 1 DFC, m¿DAB = 111°, and m/DBC = 39 degrees

What is m/DEF?

A) 30
B) 51
C) 60
D) 120

Answer 1

Let's break this down to find \( m\angle DEF \).

Given:
- \( m\angle DAB = 111^\circ \)
- \( m\angle DBC = 39^\circ \)

Considering that \( \angle DAB \) and \( \angle DBC \) are supplementary angles because they form a straight line (180 degrees), we can find \( \angle ABD \).

\( \angle DAB + \angle DBC = 111^\circ + 39^\circ = 150^\circ \)

Since \( \angle ABD \) and \( \angle DBC \) form a straight line, \( \angle ABD = 180^\circ - 150^\circ = 30^\circ \).

Now, looking at triangle \( ABD \) and triangle \( EBD \) within parallelogram \( ABCD \), we have \( \angle ABD = \angle EBD \) (alternate interior angles of parallel lines).

Therefore, \( \angle EBD = 30^\circ \).

Since \( \angle EBD \) and \( \angle DFC \) are corresponding angles (they lie on the same side of the transversal EF), \( \angle DFC = 30^\circ \).

Now, \( \angle DEF = \angle DFC = 30^\circ \).

So, the measure of \( \angle DEF \) is \( \mathbf{30^\circ} \), which matches option A.