Question

two parallel lines, (g) and (h), are cut by the transversal (k) as shown.
suppose (m angle 1 = 55^{circ}).

complete the statements below.

we see that (angle 1) and (angle 2) are vertical angles.
thus, (angle 1) and (angle 2) are congruent.
so, (m angle 2 = 55^{circ}).

we see that (angle 2) and (angle 3) are alternate interior angles.
and since the lines (g) and (h) are parallel, (angle 2) and (angle 3) are congruent.
so, (m angle 3 = 55^{circ}).

therefore, (angle 1) and (angle 3) are congruent.
we also see that (angle 1) and (angle 3) are corresponding angles.

the relationship between (angle 1) and (angle 3) is an example of the following rule.
when parallel lines are cut by a transversal, corresponding angles are congruent.

Explanation:

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

Step 1: Analyze the relationship between \(\angle 1\) and \(\angle 2\)

• \(\angle 1\) and \(\angle 2\) are opposite each other at the intersection of line \(g\) and transversal \(k\).
• Relationship: vertical angles
• Property: congruent
• Measure: \(m\angle 2 = 55^{\circ}\)

Step 2: Analyze the relationship between \(\angle 2\) and \(\angle 3\)

• \(\angle 2\) and \(\angle 3\) lie between the parallel lines \(g\) and \(h\) on opposite sides of the transversal \(k\).
• Relationship: alternate interior angles
• Property: congruent
• Measure: \(m\angle 3 = 55^{\circ}\)

Step 3: Analyze the relationship between \(\angle 1\) and \(\angle 3\)

• Since \(m\angle 1 = 55^{\circ}\) and \(m\angle 3 = 55^{\circ}\), they are congruent.
• They lie in the same relative position at each intersection (bottom-left).
• Relationship: corresponding angles

Step 4: Identify the general rule

• The relationship between \(\angle 1\) and \(\angle 3\) demonstrates that when parallel lines are cut by a transversal, corresponding angles are congruent.

Answer:

• We see that \(\angle 1\) and \(\angle 2\) are vertical angles.
• Thus, \(\angle 1\) and \(\angle 2\) are congruent.
• So, \(m\angle 2 = 55^{\circ}\).
• We see that \(\angle 2\) and \(\angle 3\) are alternate interior angles.
• And since the lines \(g\) and \(h\) are parallel, \(\angle 2\) and \(\angle 3\) are congruent.
• So, \(m\angle 3 = 55^{\circ}\).
• Therefore, \(\angle 1\) and \(\angle 3\) are congruent.
• We also see that \(\angle 1\) and \(\angle 3\) are corresponding angles.
• The relationship between \(\angle 1\) and \(\angle 3\) is an example of the following rule: When parallel lines are cut by a transversal, corresponding angles are congruent.