Question

give a pair of alternate interior angles, a pair of corresponding angles, and a pair of alternate exterior angles.

(a) alternate interior angles: \\(\angle\\) and \\(\angle\\)
(b) corresponding angles: \\(\angle\\) and \\(\angle\\)
(c) alternate exterior angles: \\(\angle\\) and \\(\angle\\)

Explanation:

⚡ Using what you learned: lines and angles (parallel, transversal, vertical)

Step 1: Identify Alternate Interior Angles

Alternate interior angles lie between the two lines \(a\) and \(b\) (interior) and on opposite sides of the transversal line \(m\) (alternate).

Looking at the interior region between lines \(a\) and \(b\):

• The interior angles are \(\angle 3\), \(\angle 4\), \(\angle 5\), and \(\angle 8\).
• Pairs on opposite sides of the transversal \(m\) are:
• \(\angle 3\) and \(\angle 6\) (Note: \(\angle 6\) is actually an interior angle on the right line \(b\). Let's trace the numbers carefully: left intersection has \(\angle 2\) top-left, \(\angle 3\) top-right, \(\angle 1\) bottom-left, \(\angle 4\) bottom-right. Right intersection has \(\angle 8\) top-left, \(\angle 7\) top-right, \(\angle 5\) bottom-left, \(\angle 6\) bottom-right).
• The interior angles between lines \(a\) and \(b\) are \(\angle 3\), \(\angle 4\) (on line \(a\)) and \(\angle 5\), \(\angle 8\) (on line \(b\)).
• Alternate interior pairs: \(\angle 3\) and \(\angle 5\) (one is top-right, the other is bottom-left relative to their intersections), or \(\angle 4\) and \(\angle 8\) (one is bottom-right, the other is top-left).

A valid pair is:

$$ \angle 3 \text{ and } \angle 5 $$

Step 2: Identify Corresponding Angles

Corresponding angles lie in the same relative position at each intersection where the transversal crosses the two lines.

Comparing the positions:

• Top-left: \(\angle 2\) and \(\angle 8\)
• Top-right: \(\angle 3\) and \(\angle 7\)
• Bottom-left: \(\angle 1\) and \(\angle 5\)
• Bottom-right: \(\angle 4\) and \(\angle 6\)

A valid pair is:

$$ \angle 2 \text{ and } \angle 8 $$

Step 3: Identify Alternate Exterior Angles

Alternate exterior angles lie outside the two lines \(a\) and \(b\) (exterior) and on opposite sides of the transversal line \(m\) (alternate).

The exterior angles are \(\angle 1\), \(\angle 2\) (on line \(a\)) and \(\angle 6\), \(\angle 7\) (on line \(b\)).

• Pairs on opposite sides of the transversal \(m\) are:
• \(\angle 2\) (top-left) and \(\angle 6\) (bottom-right)
• \(\angle 1\) (bottom-left) and \(\angle 7\) (top-right)

A valid pair is:

$$ \angle 2 \text{ and } \angle 6 $$

Answer:

• (a) Alternate interior angles: \(\angle 3\) and \(\angle 5\) (or \(\angle 4\) and \(\angle 8\))
• (b) Corresponding angles: \(\angle 2\) and \(\angle 8\) (or \(\angle 3\) and \(\angle 7\), \(\angle 1\) and \(\angle 5\), \(\angle 4\) and \(\angle 6\))
• (c) Alternate exterior angles: \(\angle 2\) and \(\angle 6\) (or \(\angle 1\) and \(\angle 7\))