Question

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

complete the statements below.

we see that (angle 1) and (angle 2) are select
thus, (angle 1) and (angle 2) are select
so, (mangle 2 = ^{circ}).

we see that (angle 2) and (angle 3) are select
and since the lines (g) and (h) are parallel, (angle 2) and (angle 3) are select
so, (mangle 3 = ^{circ}).

therefore, (angle 1) and (angle 3) are select
we also see that (angle 1) and (angle 3) are select

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

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 angles formed by 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 on opposite sides of the transversal \(k\) and between the two parallel lines \(g\) and \(h\).

• 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 \(m\angle 3 = 55^{\circ}\):

• Relationship: equal in measure (or congruent)
• Position: They lie in the same relative position at each intersection (bottom-left). Thus, they are corresponding angles.

Step 4: General rule

When parallel lines are cut by a transversal, corresponding angles are congruent (or equal).

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 equal in measure (or congruent).
• We also see that \(\angle 1\) and \(\angle 3\) are corresponding angles.
• When parallel lines are cut by a transversal, corresponding angles are congruent (or equal).