Question

practice supplementary angles. which angle pairs are supplementary? check all that apply. ∠1 and ∠2 ∠4 and ∠3 ∠4 and ∠5 ∠7 and ∠5 ∠3 and ∠6

Explanation:

Step1: Recall supplementary angles definition

Supplementary angles sum to \(180^\circ\) (a straight angle).

Step2: Analyze \(\angle1\) and \(\angle2\)

\(\angle1\) and \(\angle2\) are adjacent but form a smaller angle, not \(180^\circ\). Not supplementary.

Step3: Analyze \(\angle4\) and \(\angle3\)

\(\angle3\), \(\angle4\), \(\angle5\), \(\angle6\) are around a point. \(\angle3 + \angle4 = 180^\circ\) (straight line). Supplementary.

Step4: Analyze \(\angle4\) and \(\angle5\)

\(\angle4 + \angle5\) is less than \(180^\circ\) (they are adjacent in a triangle - like structure? No, around a point but not a straight line). Wait, no: \(\angle3\), \(\angle4\), \(\angle5\), \(\angle6\) are vertical angles? Wait, no, the intersection: \(\angle3\) and \(\angle5\) are vertical? Wait, no, let's re - look. The lines intersect, so \(\angle3 + \angle4 + \angle5 + \angle6 = 360^\circ\), but \(\angle3\) and \(\angle6\) are vertical? No, \(\angle3\) and \(\angle5\)? Wait, no, adjacent angles on a straight line: \(\angle3\) and \(\angle4\) are adjacent on a straight line, so they sum to \(180^\circ\). \(\angle4\) and \(\angle5\): wait, maybe I made a mistake. Wait, \(\angle4\) and \(\angle5\): are they adjacent? Wait, the angle \(\angle4\), \(\angle5\), and the triangle's angle. Wait, no, let's check \(\angle4\) and \(\angle5\): if we look at the intersection, \(\angle4\) and \(\angle5\) are adjacent, but do they form a straight line? No, \(\angle3\) and \(\angle4\) form a straight line, \(\angle4\) and \(\angle5\) are part of a triangle? Wait, no, the lines: the two intersecting lines create \(\angle3\), \(\angle4\), \(\angle5\), \(\angle6\). So \(\angle3\) and \(\angle6\) are vertical, \(\angle4\) and \(\angle5\) are vertical? No, \(\angle3 + \angle4 = 180^\circ\) (straight line), \(\angle4 + \angle5 = 180^\circ\)? Wait, no, if \(\angle3\) and \(\angle5\) are vertical, then \(\angle3=\angle5\), and \(\angle4=\angle6\). Wait, maybe I messed up. Let's check \(\angle4\) and \(\angle5\): if the two lines are intersecting, then adjacent angles on a straight line sum to \(180^\circ\). So \(\angle3\) and \(\angle4\) are supplementary, \(\angle4\) and \(\angle5\): wait, no, \(\angle4\) and \(\angle5\) are adjacent to a triangle? Wait, no, the angle \(\angle4\) and \(\angle5\): maybe they are supplementary? Wait, no, let's check the other options. \(\angle4\) and \(\angle5\): if we consider the straight line, maybe. Wait, let's check \(\angle4\) and \(\angle5\): sum to \(180^\circ\)? Wait, no, \(\angle3 + \angle4 + \angle5 + \angle6 = 360^\circ\), and if \(\angle3=\angle5\) and \(\angle4=\angle6\) (vertical angles), then \(\angle3+\angle4 = 180^\circ\), \(\angle4+\angle5=\angle4+\angle3 = 180^\circ\) (since \(\angle3=\angle5\)). Oh, right! So \(\angle4\) and \(\angle5\) are supplementary.

Step5: Analyze \(\angle4\) and \(\angle5\) (re - check)

Since \(\angle3\) and \(\angle5\) are vertical angles (\(\angle3=\angle5\)), and \(\angle3+\angle4 = 180^\circ\) (supplementary), then \(\angle4+\angle5=\angle4+\angle3 = 180^\circ\), so they are supplementary.

Step6: Analyze \(\angle7\) and \(\angle5\)

\(\angle7\) and \(\angle5\): are they related? \(\angle5\) and \(\angle7\): let's see, the triangle and the transversal. Wait, \(\angle5\) and \(\angle7\) are same - side interior angles? No, maybe not. Wait, no, \(\angle7\) and \(\angle5\): do they sum to \(180^\circ\)? No, probably not. Wait, no, let's check the straight line. \(\angle7\) and \(\angle8\) are supplementary, but \(\angle5\) and \(\angle7\): no.

Step7: Analyze \(\angle3\) and \(\angle6\)

\(\angle3\) and \(\angle6\) are vertical angles, so they are equal, not supplementary (unless they are \(90^\circ\), but we don't know that). So they are not supplementary.

Wait, let's re - evaluate:

- \(\angle1\) and \(\angle2\): adjacent, but form a small angle, sum less than \(180^\circ\). Not supplementary.
- \(\angle4\) and \(\angle3\): adjacent on a straight line, sum to \(180^\circ\). Supplementary.
- \(\angle4\) and \(\angle5\): as \(\angle3=\angle5\) (vertical angles) and \(\angle3+\angle4 = 180^\circ\), so \(\angle4+\angle5 = 180^\circ\). Supplementary.
- \(\angle7\) and \(\angle5\): no, they don't form a straight line.
- \(\angle3\) and \(\angle6\): vertical angles, equal, not supplementary.

Wait, maybe I made a mistake with \(\angle4\) and \(\angle5\). Let's look at the diagram again. The two intersecting lines: so \(\angle3\), \(\angle4\), \(\angle5\), \(\angle6\) are around the intersection. So \(\angle3\) and \(\angle4\) are adjacent (form a straight line), so supplementary. \(\angle4\) and \(\angle5\) are adjacent (form a straight line? Wait, no, \(\angle3\), \(\angle4\), \(\angle5\), \(\angle6\): \(\angle3\) and \(\angle6\) are vertical, \(\angle4\) and \(\angle5\) are vertical? No, \(\angle3\) and \(\angle5\) are vertical, \(\angle4\) and \(\angle6\) are vertical. So \(\angle3+\angle4 = 180^\circ\) (supplementary), \(\angle4+\angle5=\angle4+\angle3 = 180^\circ\) (since \(\angle3=\angle5\)), so yes, supplementary. \(\angle4\) and \(\angle3\): supplementary. \(\angle4\) and \(\angle5\): supplementary. Wait, also, what about \(\angle7\) and \(\angle5\)? No, \(\angle7\) and \(\angle5\): maybe \(\angle5\) and \(\angle7\) are same - side interior angles? No, the triangle: \(\angle5\) is at the intersection, \(\angle7\) is at the other intersection. Wait, maybe \(\angle7\) and \(\angle5\) are not supplementary. And \(\angle3\) and \(\angle6\) are vertical, so equal, not supplementary.

So the supplementary pairs are \(\angle4\) and \(\angle3\), \(\angle4\) and \(\angle5\). Wait, but let's check the options again:

Options:

- \(\angle1\) and \(\angle2\): No.
- \(\angle4\) and \(\angle3\): Yes.
- \(\angle4\) and \(\angle5\): Yes.
- \(\angle7\) and \(\angle5\): No.
- \(\angle3\) and \(\angle6\): No.

Wait, maybe I missed \(\angle7\) and \(\angle5\). Wait, \(\angle5\) and \(\angle7\): if we consider the transversal, maybe they are same - side interior angles? No, the triangle has angles \(\angle1\), \(\angle2\), and the angle at the intersection. Wait, \(\angle5\) is part of the intersection, \(\angle7\) is at the other intersection. Maybe \(\angle5\) and \(\angle7\) are supplementary? No, I think my initial analysis is correct.

Answer:

\(\angle4\) and \(\angle3\), \(\angle4\) and \(\angle5\) (so the options to check are \(\boldsymbol{\angle4}\) and \(\boldsymbol{\angle3}\), \(\boldsymbol{\angle4}\) and \(\boldsymbol{\angle5}\))