Question

in the diagram, m∠3 = 120° and m∠12 = 80°. which angle measures are correct? check all that apply. m∠1 = 60°; m∠13 = 80°; m∠6 = 80°; m∠5 = 60°; m∠10 = 120°; m∠14 = 100°

Explanation:

Step1: Analyze ∠1 and ∠3

∠1 and ∠3 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 ∠13 and ∠12

∠12 and ∠13 are alternate interior angles (lines \( e, f \) cut by \( d \)), so \( m\angle13 = m\angle12 = 80^\circ \). So \( m\angle13 = 80^\circ \) is correct.

Step3: Analyze ∠6 and ∠3

∠3 and ∠6: ∠3 and ∠5 are same - side interior angles? Wait, ∠3 and ∠5: lines \( e, f \) cut by \( c \). Wait, ∠3 and ∠6: ∠3 and ∠4 are supplementary, ∠4 and ∠6 are alternate interior angles? Wait, \( m\angle3 = 120^\circ \), ∠3 and ∠5: if lines \( e\parallel f \) (assuming, since there are transversals), ∠3 and ∠5 would be same - side interior angles, so \( m\angle3 + m\angle5 = 180^\circ \), so \( m\angle5 = 60^\circ \). Then ∠5 and ∠6 are supplementary? Wait, no, ∠5 and ∠6 are a linear pair? Wait, no, ∠5 and ∠6: looking at the diagram, ∠5 and ∠6 are adjacent, forming a linear pair? Wait, no, the transversal \( c \) cuts \( e \) and \( f \). ∠3 and ∠5: same - side interior angles, so if \( e\parallel f \), \( m\angle3 + m\angle5 = 180^\circ \), so \( m\angle5 = 60^\circ \). Then ∠5 and ∠6: are they vertical angles? No, ∠5 and ∠7 are vertical, ∠6 and ∠8 are vertical. Wait, ∠4 and ∠6: alternate interior angles, \( m\angle4 = 180 - 120 = 60^\circ \), so \( m\angle6 = 60^\circ \), so \( m\angle6 = 80^\circ \) is wrong.

Step4: Analyze ∠5

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

Step5: Analyze ∠10

∠10 and ∠12: ∠10 and ∠12 are same - side interior angles? Wait, ∠10 and ∠11 are supplementary, ∠11 and ∠13 are alternate interior angles (\( m\angle13 = 80^\circ \)), so \( m\angle11 = 80^\circ \), then \( m\angle10 = 180 - 80 = 100^\circ \), so \( m\angle10 = 120^\circ \) is wrong.

Step6: Analyze ∠14

∠12 and ∠14 are supplementary (linear pair), so \( m\angle14 = 180 - 80 = 100^\circ \). So \( m\angle14 = 100^\circ \) is correct.

Answer:

• \( m\angle1 = 60^\circ \)
• \( m\angle13 = 80^\circ \)
• \( m\angle5 = 60^\circ \)
• \( m\angle14 = 100^\circ \)