Question

which angles are complementary to ∠5? select all that apply.
∠6 ∠4 ∠1 ∠3

Explanation:

Step1: Recall Complementary Angles

Complementary angles sum to \(90^\circ\). From the diagram, \(\angle 4\) and \(\angle 6\) are right angles (marked with a square), so \(\angle 4 = \angle 6 = 90^\circ\).

Step2: Analyze \(\angle 5\) and \(\angle 4\)

\(\angle 5 + \angle 4=\angle 6\) (since \(\angle 6 = 90^\circ\)), so \(\angle 5 + \angle 4 = 90^\circ\). Thus, \(\angle 4\) is complementary to \(\angle 5\).

Step3: Analyze \(\angle 5\) and \(\angle 1\)

Vertical angles and linear pairs: \(\angle 1\) and \(\angle 4\) are vertical? Wait, no. Wait, \(\angle 1\) and \(\angle 3\) are vertical, \(\angle 4\) and \(\angle 1\)? Wait, actually, \(\angle 5\) and \(\angle 1\): since \(\angle 5 + \angle 4 = 90^\circ\) and \(\angle 4=\angle 1\) (vertical angles? Wait, no, \(\angle 4\) and \(\angle 1\) are adjacent to the right angle. Wait, \(\angle 4\) and \(\angle 1\) are equal? Wait, no, \(\angle 4\) is part of the right angle, \(\angle 1\) is equal to \(\angle 4\) because they are vertical? Wait, no, the horizontal and vertical lines are perpendicular, so \(\angle 4 = \angle 1 = 90^\circ - \angle 5\)? Wait, no, \(\angle 5 + \angle 4 = 90^\circ\) (since \(\angle 6 = 90^\circ\)), and \(\angle 1=\angle 4\) (vertical angles), so \(\angle 5 + \angle 1 = 90^\circ\). Thus, \(\angle 1\) is also complementary. Wait, but let's check again.

Wait, the right angle is \(\angle 4 + \angle 5 = 90^\circ\) (since \(\angle 6\) is a right angle). Also, \(\angle 1\) and \(\angle 4\) are equal (vertical angles? Wait, no, \(\angle 4\) and \(\angle 1\) are adjacent to the intersection. Wait, actually, \(\angle 4\) and \(\angle 1\) are congruent because they are vertical angles? Wait, no, the horizontal line and vertical line intersect, so \(\angle 4\) and \(\angle 1\) are both equal to \(90^\circ - \angle 5\)? Wait, no, \(\angle 4 + \angle 5 = 90^\circ\) (because \(\angle 6 = 90^\circ\)), and \(\angle 1 = \angle 4\) (vertical angles), so \(\angle 5 + \angle 1 = 90^\circ\). So \(\angle 4\) and \(\angle 1\) are complementary to \(\angle 5\). Wait, but the options are \(\angle 6\), \(\angle 4\), \(\angle 1\), \(\angle 3\). \(\angle 6\) is \(90^\circ\), so \(\angle 5 + \angle 6 = \angle 5 + 90^\circ>90^\circ\), so not complementary. \(\angle 3\) is equal to \(\angle 1\) (vertical angles), but \(\angle 3\) is \(90^\circ + \angle 2\), so no. So the correct ones are \(\angle 4\) and \(\angle 1\). Wait, but let's re-express:

From the diagram, the vertical line and the slanted line form \(\angle 5\). The horizontal and vertical lines are perpendicular, so \(\angle 4 + \angle 5 = 90^\circ\) (since \(\angle 6 = 90^\circ\)). Also, \(\angle 1 = \angle 4\) (vertical angles), so \(\angle 5 + \angle 1 = 90^\circ\). Thus, \(\angle 4\) and \(\angle 1\) are complementary to \(\angle 5\).

Answer:

\(\angle 4\), \(\angle 1\)