Question

1. which is a correct two - column proof for the figure and information?

given ( r parallel s )
prove: ( angle a ) and ( angle g ) are supplementary
a.

statements	reasons
2. ( angle a cong angle d )	2. vertical angle thm.
3. ( angle d ) and ( angle f ) are supplementary.	3. same - side interior angles post.
4. ( angle f cong angle g )	4. vertical angle thm.
5. ( angle a ) and ( angle g ) are supplementary.	5. substitution

b.

statements	reasons
2. ( angle a cong angle d )	2. vertical angles thm.
3. ( angle d ) and ( angle g ) are supplementary.	3. alternate interior angles thm.
4. ( angle f cong angle g )	4. vertical angles thm.
5. ( angle a ) and ( angle g ) are supplementary.	5. same - side interior angles post.

c.

statements	reasons
2. ( angle a cong angle d )	2. alternate interior angles thm.
3. ( angle d ) and ( angle f ) are supplementary.	3. same - side interior angles post.
4. ( angle f cong angle g )	4. corresponding angles thm.
5. ( angle a ) and ( angle g ) are supplementary.	5. substitution

Explanation:

Step1: Check vertical angles

Vertical angles are congruent. So, \(\angle a\cong\angle d\) (Vertical Angle Thm).

Step2: Check same - side interior angles

Since \(r\parallel s\), \(\angle d\) and \(\angle f\) are same - side interior angles. By the Same - Side Interior Angles Postulate, \(\angle d\) and \(\angle f\) are supplementary.

Step3: Check vertical angles again

\(\angle f\cong\angle g\) (Vertical Angle Thm).

Step4: Substitute

Since \(\angle a\cong\angle d\) and \(\angle f\cong\angle g\), and \(\angle d\) and \(\angle f\) are supplementary, by substitution, \(\angle a\) and \(\angle g\) are supplementary.

Now let's check each option:

• Option A:
• Step 1: \(r\parallel s\) (Given)
• Step 2: \(\angle a\cong\angle d\) (Vertical Angle Thm)
• Step 3: \(\angle d\) and \(\angle f\) are supplementary (Same - Side Interior Angles Post)
• Step 4: \(\angle f\cong\angle g\) (Vertical Angle Thm)
• Step 5: \(\angle a\) and \(\angle g\) are supplementary (Substitution). This is correct.
• Option B:
• Step 3: \(\angle d\) and \(\angle g\) are supplementary. The reason "Alternate Interior Angles Thm" is wrong. Alternate Interior Angles Thm is about congruence of angles, not supplementary. Also, step 5 reason is wrong.
• Option C:
• Step 2: \(\angle a\cong\angle d\) is by Vertical Angle Thm, not Alternate Interior Angles Thm. Step 4: \(\angle f\cong\angle g\) is by Vertical Angle Thm, not Corresponding Angles Thm.

Answer:

A.

1. \(r\parallel s\) - Given
2. \(\angle a\cong\angle d\) - Vertical Angle Thm
3. \(\angle d\) and \(\angle f\) are supplementary - Same - Side Interior Angles Post
4. \(\angle f\cong\angle g\) - Vertical Angle Thm
5. \(\angle a\) and \(\angle g\) are supplementary - Substitution