Question

give a pair of corresponding angles, a pair of alternate exterior angles, and a pair of alternate interior angles. (a) corresponding angles: ∠ and ∠ (b) alternate exterior angles: ∠ and ∠ (c) alternate interior angles: ∠ and ∠

Explanation:

Step1: Recall angle definitions

- Corresponding angles: Same position relative to parallel lines and transversal. E.g., $\angle1$ and $\angle3$, $\angle5$ and $\angle7$, $\angle2$ and $\angle4$, $\angle6$ and $\angle8$.
- Alternate exterior angles: Outside parallel lines, opposite sides of transversal. E.g., $\angle1$ and $\angle8$, $\angle2$ and $\angle7$.
- Alternate interior angles: Inside parallel lines, opposite sides of transversal. E.g., $\angle2$ and $\angle7$ (wait, no, $\angle6$ and $\angle3$, $\angle2$ and $\angle7$? Wait, correct: $\angle2$ and $\angle7$? No, $\angle6$ and $\angle3$, $\angle2$ and $\angle7$? Wait, parallel lines \(a\) and \(b\), transversal \(n\). Interior angles: between \(a\) and \(b\), so $\angle2$, $\angle6$, $\angle3$, $\angle7$? Wait, no: \(a\) and \(b\) are parallel, transversal \(n\). So:
- Corresponding: $\angle1$ (above \(a\), left of \(n\)) and $\angle3$ (above \(b\), left of \(n\)); $\angle5$ (above \(a\), right of \(n\)) and $\angle7$ (above \(b\), right of \(n\)); $\angle2$ (below \(a\), left of \(n\)) and $\angle4$ (below \(b\), left of \(n\)); $\angle6$ (below \(a\), right of \(n\)) and $\angle8$ (below \(b\), right of \(n\)).
- Alternate exterior: $\angle1$ (exterior to \(a - b\) area, left) and $\angle8$ (exterior, right); $\angle2$ (exterior, left) and $\angle7$ (exterior, right).
- Alternate interior: $\angle2$ (interior, left) and $\angle7$ (interior, right)? No, wait: interior is between \(a\) and \(b\), so $\angle6$ (below \(a\), right) and $\angle3$ (below \(b\), left); $\angle2$ (below \(a\), left) and $\angle7$ (below \(b\), right)? Wait, no, correct alternate interior: $\angle6$ and $\angle3$, $\angle2$ and $\angle7$? Wait, no, $\angle2$ is below \(a\), left of \(n\); $\angle7$ is below \(b\), right of \(n\) – no, that's not alternate. Wait, alternate interior: angles between the two lines, on opposite sides of transversal. So between \(a\) and \(b\): $\angle2$ (below \(a\), left of \(n\)), $\angle6$ (below \(a\), right of \(n\)), $\angle3$ (above \(b\), left of \(n\)), $\angle7$ (above \(b\), right of \(n\))? Wait, no, \(a\) and \(b\) are horizontal, \(n\) is slanting. So:
- Corresponding: $\angle1$ (top - left of \(n\) relative to \(a\)) and $\angle3$ (top - left of \(n\) relative to \(b\)); $\angle5$ (top - right) and $\angle7$ (top - right); $\angle2$ (bottom - left) and $\angle4$ (bottom - left); $\angle6$ (bottom - right) and $\angle8$ (bottom - right).
- Alternate exterior: $\angle1$ (outside \(a - b\) area, top - left) and $\angle8$ (outside, bottom - right); $\angle2$ (outside, bottom - left) and $\angle7$ (outside, top - right).
- Alternate interior: $\angle2$ (inside \(a - b\), bottom - left) and $\angle7$ (inside, top - right)? No, wait, inside \(a - b\) is between \(a\) and \(b\), so $\angle6$ (inside, bottom - right) and $\angle3$ (inside, top - left); $\angle2$ (inside, bottom - left) and $\angle7$ (inside, top - right)? No, that's not alternate. Wait, correct: alternate interior angles are $\angle6$ and $\angle3$, $\angle2$ and $\angle7$? Wait, no, $\angle6$ is below \(a\), right of \(n\); $\angle3$ is above \(b\), left of \(n\) – no, opposite sides of transversal. Transversal \(n\) splits into left and right. So $\angle6$ (right of \(n\), between \(a\) and \(b\)) and $\angle3$ (left of \(n\), between \(a\) and \(b\)) – yes, alternate interior. Similarly, $\angle2$ (left of \(n\), between \(a\) and \(b\)) and $\angle7$ (right of \(n\), between \(a\) and \(b\)) – yes, alternate interior.

Step2: Choose examples

- (a) Corresponding: $\angle1$ and $\angle3$ (or any other corresponding pair).
- (b) Alternate exterior: $\angle1$ and $\angle8$ (or $\angle2$ and $\angle7$).
- (c) Alternate interior: $\angle2$ and $\angle7$ (or $\angle6$ and $\angle3$).

Answer:

(a) Corresponding angles: $\angle1$ and $\angle3$ (or $\angle5$ and $\angle7$, $\angle2$ and $\angle4$, $\angle6$ and $\angle8$)
(b) Alternate exterior angles: $\angle1$ and $\angle8$ (or $\angle2$ and $\angle7$)
(c) Alternate interior angles: $\angle2$ and $\angle7$ (or $\angle6$ and $\angle3$)

(Note: Answers may vary as long as they fit the definitions.)