Question

what is the relationship between $\angle m$ and $\angle n$?
choose 1 answer:
a vertical angles
b complementary angles
c supplementary angles
d none of the above

Explanation:

To determine the relationship between \( \angle m \) and \( \angle n \), we analyze the definitions:

• Vertical angles are opposite angles formed by intersecting lines, but \( \angle m \) and \( \angle n \) are adjacent, so A is incorrect.
• Complementary angles sum to \( 90^\circ \). From the diagram, \( \angle DAF \) is a straight angle ( \( 180^\circ \))? No, wait, \( \angle DAF \) is actually a right angle? Wait, looking at the diagram, lines \( DE \) and \( CF \) intersect at \( A \), and \( DA \) and \( FA \) are a straight line (vertical line), so \( \angle DAF = 180^\circ \)? No, wait, \( DA \) and \( FA \) are opposite rays, so \( \angle DAF = 180^\circ \)? Wait, no, in the diagram, \( DA \) is up, \( FA \) is down, so they are a straight line ( \( 180^\circ \)). But \( \angle m \) and \( \angle n \) are adjacent angles with \( \angle DAB \) and \( \angle BAC \)? Wait, no, the key is: if \( DA \) and \( FA \) are perpendicular to \( EC \)? Wait, no, the diagram shows that \( \angle m + \angle n = 90^\circ \)? Wait, no, maybe \( DA \) and \( FA \) are a straight line, but \( \angle m \) and \( \angle n \) are adjacent and form a right angle? Wait, no, let's re-express: the angle between \( DA \) and \( FA \) is \( 180^\circ \), but \( \angle m \) and \( \angle n \) are adjacent angles that add up to \( 90^\circ \) (since \( DA \) and \( FA \) are perpendicular? Wait, maybe the diagram has \( DA \perp EC \)? No, the diagram shows \( E \)---\( A \)---\( C \) as a horizontal line, and \( D \)---\( A \)---\( F \) as a vertical line, so \( \angle DAE = 90^\circ \)? Wait, no, \( E \)---\( A \)---\( C \) is horizontal, \( D \)---\( A \)---\( F \) is vertical, so they are perpendicular, meaning \( \angle DAC = 90^\circ \)? Wait, no, \( E \)---\( A \)---\( C \) is horizontal, \( D \)---\( A \)---\( F \) is vertical, so \( \angle DAE = 90^\circ \), \( \angle FAC = 90^\circ \). Then \( \angle m \) and \( \angle n \) are adjacent angles within the right angle at \( A \) (between \( DA \) and \( EC \))? Wait, no, \( \angle m \) is between \( DA \) and \( AB \), \( \angle n \) is between \( AB \) and \( AC \). Since \( DA \) and \( AC \) are perpendicular (because \( DA \) is vertical and \( AC \) is horizontal), \( \angle DAC = 90^\circ \), so \( \angle m + \angle n = 90^\circ \), which means they are complementary. Wait, but earlier I thought \( DA \) and \( FA \) are vertical, but maybe \( DA \) and \( EC \) are perpendicular. Let's confirm the definitions:
• Complementary angles: two angles whose sum is \( 90^\circ \).
• Supplementary angles: two angles whose sum is \( 180^\circ \).
• Vertical angles: opposite angles when two lines intersect.

From the diagram, \( \angle m \) and \( \angle n \) are adjacent angles that form a right angle (since \( DA \) is vertical and \( EC \) is horizontal, so they are perpendicular, making \( \angle DAC = 90^\circ \), so \( \angle m + \angle n = 90^\circ \)). Therefore, they are complementary angles.

Wait, but let's check the options:

• A: Vertical angles: no, they are adjacent.
• B: Complementary angles: sum to \( 90^\circ \), which matches.
• C: Supplementary angles: sum to \( 180^\circ \), no.
• D: None of the above: no, since B is correct.

So the correct answer is B.

Answer:

B. Complementary angles