Question

determining angle measures

in the diagram, ( mangle 3 = 120^circ ) and ( mangle 12 = 80^circ ). which angle measures are correct? check all that apply.

( square mangle 1 = 60^circ )

( square mangle 13 = 80^circ )

( square mangle 6 = 80^circ )

( square mangle 5 = 60^circ )

( square mangle 10 = 120^circ )

( square mangle 14 = 100^circ )

Explanation:

Step1: Analyze \( m\angle1 \)

\( \angle1 \) and \( \angle3 \) are supplementary (linear pair), so \( m\angle1 + m\angle3 = 180^\circ \). Given \( m\angle3 = 120^\circ \), then \( m\angle1 = 180^\circ - 120^\circ = 60^\circ \). So \( m\angle1 = 60^\circ \) is correct.

Step2: Analyze \( m\angle13 \)

\( \angle12 \) and \( \angle13 \) are alternate interior angles (lines \( e \) and \( f \) parallel, transversal \( d \)), so \( m\angle13 = m\angle12 = 80^\circ \). So \( m\angle13 = 80^\circ \) is correct.

Step3: Analyze \( m\angle6 \)

\( \angle3 \) and \( \angle6 \) are alternate interior angles (lines \( e \) and \( f \) parallel, transversal \( c \)), so \( m\angle6 = 180^\circ - m\angle3 \)? Wait, no, \( \angle3 \) and \( \angle5 \) are same - side interior? Wait, \( \angle3 \) and \( \angle6 \): actually, \( \angle3 \) and \( \angle5 \) are same - side interior, \( \angle3 + \angle5 = 180^\circ \), but \( \angle6 \) and \( \angle12 \): \( \angle6 \) and \( \angle12 \) are corresponding angles? Wait, lines \( e \) and \( f \) are parallel, transversal \( d \): no, transversal \( c \) and \( d \). Wait, \( \angle12 = 80^\circ \), \( \angle6 \) and \( \angle12 \): are they corresponding? Wait, \( \angle6 \) and \( \angle12 \): let's see, \( \angle6 \) and \( \angle12 \): if we consider transversal \( d \), no, transversal \( c \) and \( d \). Wait, \( \angle12 = 80^\circ \), \( \angle6 \): \( \angle3 = 120^\circ \), \( \angle3 \) and \( \angle6 \): since \( e\parallel f \), \( \angle3 + \angle5 = 180^\circ \), \( \angle5 = 60^\circ \), and \( \angle6 \) and \( \angle5 \) are supplementary? No, \( \angle5 \) and \( \angle6 \) are linear pair? Wait, no, \( \angle5 \) and \( \angle6 \) are adjacent, forming a linear pair? Wait, no, the two horizontal lines are \( e \) (top) and \( f \) (bottom). The transversal \( c \) intersects \( e \) at \( \angle1,\angle2,\angle3,\angle4 \) and \( f \) at \( \angle5,\angle6,\angle7,\angle8 \). So \( \angle3 \) and \( \angle5 \) are same - side interior angles, so \( m\angle3 + m\angle5 = 180^\circ \), so \( m\angle5 = 60^\circ \). \( \angle5 \) and \( \angle6 \) are linear pair, so \( m\angle5 + m\angle6 = 180^\circ \), so \( m\angle6 = 120^\circ \), so \( m\angle6 = 80^\circ \) is wrong.

Step4: Analyze \( m\angle5 \)

As above, \( \angle3 \) and \( \angle5 \) are same - side interior angles (\( e\parallel f \), transversal \( c \)), so \( m\angle3 + m\angle5 = 180^\circ \). Given \( m\angle3 = 120^\circ \), then \( m\angle5 = 180^\circ - 120^\circ = 60^\circ \). So \( m\angle5 = 60^\circ \) is correct.

Step5: Analyze \( m\angle10 \)

\( \angle3 = 120^\circ \), \( \angle10 \) and \( \angle3 \): \( \angle10 \) and \( \angle12 \) are supplementary (linear pair), \( m\angle12 = 80^\circ \), so \( m\angle10 = 180^\circ - 80^\circ = 100^\circ \), so \( m\angle10 = 120^\circ \) is wrong.

Step6: Analyze \( m\angle14 \)

\( \angle12 = 80^\circ \), \( \angle14 \) and \( \angle12 \) are supplementary (linear pair), so \( m\angle14 = 180^\circ - 80^\circ = 100^\circ \). So \( m\angle14 = 100^\circ \) is correct.

Answer:

The correct angle measures are \( m\angle1 = 60^\circ \), \( m\angle13 = 80^\circ \), \( m\angle5 = 60^\circ \), \( m\angle14 = 100^\circ \) (i.e., the options: \( m\angle1 = 60^\circ \), \( m\angle13 = 80^\circ \), \( m\angle5 = 60^\circ \), \( m\angle14 = 100^\circ \)).