Question

in the diagram, the lines ( ell ) and ( m ) appear to be parallel and ( x = 11 ). explain why ( ell parallel m ).
(type integers or decimals. use not a number in your answers.)
a. since ( mangle bac = square ) and ( mangle efa = square ), alternate exterior angles are congruent. by the converse of the alternate exterior angles theorem, ( ell parallel m ).

b. since ( mangle bca = square ) and ( mangle efa = square ), alternate interior angles are congruent. by the converse of the alternate interior angles theorem, ( ell parallel m ).

c. since ( mangle dab = square ) and ( mangle efa = square ), corresponding angles are congruent. by the converse of the corresponding angles theorem, ( ell parallel m ).

Explanation:

Step1: Calculate m∠BAC

Given \( x = 11 \), substitute into \( m\angle BAC=(3x + 8)^\circ \).
\( m\angle BAC = 3\times11 + 8 = 33 + 8 = 41^\circ \)

Step2: Calculate m∠EFA

We know that \( m\angle EFA \) and the \( 139^\circ \) angle are supplementary (linear pair). So \( m\angle EFA = 180^\circ - 139^\circ = 41^\circ \)

Step3: Analyze the angles for option A

Since \( m\angle BAC = 41^\circ \) and \( m\angle EFA = 41^\circ \), alternate exterior angles are congruent. By the Converse of the Alternate Exterior Angles Theorem, \( l\parallel m \).

For option B:

Step1: Calculate m∠BCA

Given \( x = 11 \), substitute into \( m\angle BCA=(3x)^\circ \).
\( m\angle BCA = 3\times11 = 33^\circ \)

Step2: Calculate m∠EFA (already found as \( 41^\circ \))

\( 33^\circ
eq41^\circ \), so alternate interior angles are not congruent. So option B is incorrect.

For option C:

Step1: Calculate m∠DAB

First, find the measure of \( \angle ABC=(9x + 7)^\circ \), substitute \( x = 11 \): \( 9\times11 + 7 = 99 + 7 = 106^\circ \). Then, in triangle \( ABC \), \( m\angle DAB = m\angle ABC + m\angle BCA \) (exterior angle theorem). \( m\angle BCA = 33^\circ \), so \( m\angle DAB = 106^\circ + 33^\circ = 139^\circ \)

Step2: Compare with m∠EFA

\( m\angle EFA = 41^\circ \), \( 139^\circ
eq41^\circ \), so corresponding angles are not congruent. So option C is incorrect.

Answer:

A. Since \( m\angle BAC = \boldsymbol{41^\circ} \) and \( m\angle EFA = \boldsymbol{41^\circ} \), alternate exterior angles are congruent. By the Converse of the Alternate Exterior Angles Theorem, \( l\parallel m \).