Use the image to answer the question.

Question

Two parallel horizontal lines a and b, intersecting with two parallel slanted lines e and f. A parallelogram is formed in the center of the intersecting lines.

In the diagram, a∥b
and e∥f
. Sylvia writes a proof to prove that opposite angles, ∠6
and ∠11
, are congruent in the parallelogram. Drag and drop the statements and reasons into their correct locations in the two-column proof.

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Statement Reason
1. a∥b
and e∥f
given
2. Response area Corresponding Angles Theorem
3. m∠5+m∠6=180°
and m∠9+m∠11=180°
definition of linear pair
4. m∠13+m∠6=180°
and m∠13+m∠11=180°
Response area
5. Response area Transitive Property of Equality
6. m∠6=m∠11
Response area

The option "Substitution Property of Equality" (6 of 6) has been selected. Press tab to choose a response area, and spacebar to insert it. Press escape to cancel.

Answer 1

To complete the two-column proof for proving that opposite angles \( \angle 6 \) and \( \angle 11 \) are congruent in the parallelogram, we will use the given statements and reasons. Here’s how each statement pairs with the correct reason:

| Statement | Reason | |-----------------------------------------------------|---------------------------------------| | 1. \( a \parallel b \) and \( e \parallel f \) | Given | | 2. \( m\angle 5 + m\angle 6 = 180^\circ \) and \( m\angle 9 + m\angle 11 = 180^\circ \) | Definition of linear pair | | 3. \( m\angle 13 + m\angle 6 = 180^\circ \) and \( m\angle 13 + m\angle 11 = 180^\circ \) | Corresponding Angles Theorem | | 4. \( m\angle 6 = m\angle 11 \) | Substitution Property of Equality | | 5. \( m\angle 6 = m\angle 11 \) | Transitive Property of Equality |

Correct placements:

• Response area (2): Corresponding Angles Theorem
• Response area (4): Substitution Property of Equality
• Response area (6): \( m\angle 6 = m\angle 11 \)

So the completed entries should look like this:

| Statement | Reason | |-----------------------------------------------------|---------------------------------------| | 1. \( a \parallel b \) and \( e \parallel f \) | Given | | 2. \( m\angle 5 + m\angle 6 = 180^\circ \) and \( m\angle 9 + m\angle 11 = 180^\circ \) | Definition of linear pair | | 3. \( m\angle 13 + m\angle 6 = 180^\circ \) and \( m\angle 13 + m\angle 11 = 180^\circ \) | Corresponding Angles Theorem | | 4. \( m\angle 6 = m\angle 11 \) | Substitution Property of Equality | | 5. \( m\angle 6 = m\angle 11 \) | Transitive Property of Equality |

It's important to ensure to follow the logical progression to show that the opposite angles of the parallelogram are congruent.