Question

use the given information to prove that ( r parallel t ).
you are not allowed to use the transitive property for parallel lines.
(you may use the transitive property for \=\ or \cong\, though.)

given: ( s parallel t ), ( angle 6 cong angle 2 )
prove: ( r parallel t )

Explanation:

Step1: Identify Given Information

Given \( s \parallel t \) and \( \angle 6 \cong \angle 2 \). We need to prove \( r \parallel t \). First, recall properties of parallel lines (alternate interior, corresponding angles, etc.).

Step2: Use \( s \parallel t \) for Angle Relationships

Since \( s \parallel t \), corresponding angles or alternate interior angles are congruent. Also, \( \angle 6 \cong \angle 2 \) (given). Let's relate \( \angle 2 \) and angles formed by transversal with \( r \) and \( s \). \( \angle 2 \) and \( \angle 4 \): if we consider transversal intersecting \( r \) and \( s \), \( \angle 2 \) and \( \angle 4 \) are vertical angles? Wait, no, \( \angle 6 \cong \angle 2 \), and \( \angle 6 \) and \( \angle 4 \): \( \angle 4 \) and \( \angle 6 \) are vertical angles? Wait, \( \angle 4 \) and \( \angle 6 \) are vertical angles? No, \( \angle 3 \) and \( \angle 5 \), \( \angle 4 \) and \( \angle 6 \) are vertical angles? Wait, \( \angle 4 \) and \( \angle 6 \) are vertical angles (opposite angles when two lines intersect), so \( \angle 4 \cong \angle 6 \) (Vertical Angles Theorem). But we know \( \angle 6 \cong \angle 2 \) (given), so by Transitive Property, \( \angle 4 \cong \angle 2 \). Now, \( \angle 2 \) and \( \angle 4 \) are alternate interior angles for lines \( r \) and \( s \) with transversal. So if alternate interior angles are congruent, \( r \parallel s \) (Alternate Interior Angles Theorem). But we need \( r \parallel t \), and we know \( s \parallel t \) (given). Wait, but we can't use transitive for parallel lines. Wait, maybe another approach. Wait, \( \angle 6 \cong \angle 2 \), and \( \angle 6 \) and \( \angle 8 \): since \( s \parallel t \), \( \angle 6 \cong \angle 8 \) (corresponding angles). Then \( \angle 2 \cong \angle 8 \) (Transitive, since \( \angle 6 \cong \angle 2 \) and \( \angle 6 \cong \angle 8 \)). \( \angle 2 \) and \( \angle 8 \) are corresponding angles for lines \( r \) and \( t \) with transversal. So if corresponding angles are congruent, \( r \parallel t \) (Corresponding Angles Postulate).
Let's structure the proof:

1. \( s \parallel t \) (Given)
2. \( \angle 6 \cong \angle 8 \) (Corresponding Angles Postulate, since \( s \parallel t \) and transversal)
3. \( \angle 6 \cong \angle 2 \) (Given)
4. \( \angle 2 \cong \angle 8 \) (Transitive Property of Congruence, from steps 2 and 3)
5. \( r \parallel t \) (Corresponding Angles Postulate, since \( \angle 2 \cong \angle 8 \) and they are corresponding angles for \( r \) and \( t \) with transversal)

Wait, but the first step in the proof table: step 1 should be " \( s \parallel t \) " with reason "Given". Then step 2: " \( \angle 6 \cong \angle 8 \) " with reason "Corresponding Angles Postulate (since \( s \parallel t \))", step 3: " \( \angle 6 \cong \angle 2 \) " (Given), step 4: " \( \angle 2 \cong \angle 8 \) " (Transitive Property), step 5: " \( r \parallel t \) " (Corresponding Angles Postulate). But the initial step in the table is step 1, so first statement is " \( s \parallel t \) " with reason "Given".

Answer:

To prove \( r \parallel t \):

Step 1:

Statement: \( s \parallel t \)
Reason: Given

(Continue with subsequent steps as above, but the first step's statement is \( s \parallel t \) with reason "Given".)

For the first step in the proof table, the statement is \( s \parallel t \) (from given information) and the reason is "Given".